ACFI-T21-06

Causality and

John F. Donoghue1∗ and Gabriel Menezes2† 1Department of Physics, University of Massachusetts, Amherst, MA 01003, USA 2Departamento de F´ısica, Universidade Federal Rural do Rio de Janeiro, 23897-000, Serop´edica, RJ, Brazil

We show how uncertainty in the of field theory is essentially inevitable when one includes . This includes the fact that lightcones are ill-defined in such a theory - independent of the UV completion of the theory. We include details of the uncertainty which arises in theories of quadratic gravity.

PACS numbers:

INTRODUCTION

In our classical world, causality is taken to mean that there is no effect before its cause. Technically, we therefore use retarded propagators, which vanish for propagation backwards-in-time, to describe classical physics. Quantum physics is a bit different. Using Feynman boundary conditions, propagators do have a backwards-in-time propagation for so-called negative energy modes1. Indeed this is required [1] in order to obtain a more general definition of causality - that field operators commute for spacelike separations [2]. When combined with a definition of past and future [3, 4], this guarantees that events are only influenced by features within their past lightcone. The commutating of field operators is easy to prove for free fields, although harder to prove for interacting theories. The result of this definition of causality is reflected in the analyticity properties of scattering amplitudes, including relations between the Euclidean and Minkowski amplitudes. In gravitational quantum field theory, causality is a more difficult concept because is not the fixed rigid structure of Minkowski QFTs. Even the idea of a particle becomes difficult to define globally. Analyticity properties no longer hold as Euclidean and Lorentzian versions of non-trivial dynamical are different. At a handwaving level, we would also expect that fluctuations in the structure of spacetime would lead to violations or at least uncertainties in the causal of field theories of gravity. This heuristic argument is part of our motivation to see what explicit calculations say about causality in the presence of quantum gravity effects. That is the goal of this paper. Because of the limitations of understanding of quantum field theory in general curved spacetimes, we are limited somewhat to perturbative calculations relatively close to flat space. For these situations, the effective field theory of gravity is useful, and we can study extended quantum field theories which attempt to go beyond the effective field theory. However, even in this limited setting we can see that there will be an inherent uncertainty in our usual concepts of causality in a gravitational setting. Perhaps the effects will be more than just uncertainty, leading to actual violations at short distance scales. We discuss the calculations which lead to these conclusions below.

OVERVIEW OF CAUSALITY IN arXiv:2106.05912v1 [hep-th] 10 Jun 2021

For a free field, it is easy to use canonical quantization to calculate the commutator of two fields

Z d3k h i [φ(x), φ(y)] = e−ik·(x−y) − e+ik·(x−y) . (1) (2π)32Ek

R 3 0 Because the integration d k/(2π)32Ek is Lorentz invariant, when (x − y) < 0 (i.e. spacelike in our metric) we can transform this to a frame where (x0 − y0) = 0. There the equal time commutator vanishes, as can be readily seen by changing the spatial vector from ~k to −~k in the second term.

1 These are antiparticle modes for fermions or charged particles, but are their own antiparticle so the modes are best categorized as negative energy. 2

However this does not imply that all propagation is forward in time. The Feynman propagator Z d4k i D (x − y) = h0|T φ(x)φ(y)|0i = e−ik·(x−y) (2) F (2π)4 k2 − m2 + i can be evaluated for x0 − y0 < 0, with the result Z d3q i(Eq t−~q·~x) DF (x − y) = 3 e when x0 − y0 < 0 . (3) (2π) 2Eq We have discussed elsewhere in more detail how the factor of i determines the arrow of causality, and how it emerges from the factors of i in the quantization procedure[3, 4]. In its simplest and briefest form, it can be seen in the path integral quantization Z Z[J] = [dφ]e+iS(φ,J) Z +i R d4x[ 1 (∂ φ∂µφ−m2φ2)+Jφ] = [dφ]e 2 µ . (4) using eiS rather than e−iS. In path integral quantization, this is the only place where i enters the theory. The operation of time reversal T is antiunitary, changing the sign of i and reversing the direction of causality. In a gravitational theory, there is no global timelike direction in a generic background. Nor are there global Lorentz boosts, such as employed in the proof of the vanishing of the commutator. Nevertheless, classically there are well-defined light cones, and attempts to construct propagators on a classical background hypothesize that the free field commutators will still vanish outside of the backwards classical lightcone. While this is plausible, it becomes a daunting requirement if the spacetime is itself quantum mechanically dynamical. There is a particular UV completion of gravity which we find useful in discussing causality. This is quadratic gravity [5–10] with the Z   4 √ 2 √ 1 2 1 µναβ Squad = d x −g 2 −gR + 2 R − 2 CµναβC (5) κ 6f0 2ξ

2 where κ = 32πG and f0, ξ are coupling constants. Cµναβ is the Weyl tensor, where 1 1  1  1  − C Cµναβ = − R Rµν − R2 + G . (6) 2ξ2 µναβ ξ2 µν 3 2 This is a renormalizable field theory. One of its special features is an unstable high mass ghost in the spin-two channel. To see this, we note that the higher curvature terms involve four derivatives and lead to quartic terms in the propagator. In the spin-two channel, including the vacuum polarization diagram of the matter fields, this leads to

(2) iDµναβ = iPµναβD2(q) κ2q4 κ2q4N −q2 − i D−1(q) = q2 + i − − eff ln (7) 2 2ξ2(µ) 640π2 µ2

(2) Here, Pµναβ is the spin-two projector 1 1 P(2) = (θ θ + θ θ ) − θ θ , (8) µνρσ 2 µρ νσ µσ νρ 3 µν ρσ 2 2 and Neff are effective number of light degrees of freedom . Besides the usual pole at q = 0 there is a high mass pole and near that pole the propagator has the form −i iD (q) ∼ (9) 2 q2 − m2 − iγ

2 There are also some logarithmic contributions which do not contribute to imaginary parts in the propagator [8] which are not displayed here. 3 where γ > 0 follows from the factor of i in the logarithm. As far as causality is concerned this has two relevant features - the sign in the numerator and the sign of the imaginary part in the denominator. Both of these are changes of i to -−i and indicate that this corresponds to time-reversed propagation. More detailed analysis of the propagators confirms this [3]. We have called this resonance a Merlin mode because it propagates backwards in time compared to the usual modes. Quadratic gravity is of course only one of the possible UV completions of gravity. However because it forms a field theory, and because we can easily match on to the effective field theory, we find it a useful example for our discussion. Moreover, as it is in some ways the most conservative UV completion, it may in be relevant to Planck scale phenomenology.

THE LIGHTCONE IS NOT A QUANTUM GRAVITY CONCEPT

Many discussions of quantum gravity use concepts which are carried over from classical gravity, such as lightcones, and Penrose diagrams. However, these are not well defined in quantum gravity. A necessary consequence is that causality becomes a fuzzy concept in gravitational backgrounds, even if they are asymptotically flat. The problem with lightcones can be illustrated even within the effective field theory limit of gravity. Classically any massless particle defines a lightcone. Since the speed of propagation is the same for all massless particles, these lightcones would be identical for all forms of massless particles. However in a gravitational background this is no longer true. As an example the gravitational bending angle of a massless particle around a heavy mass has been calculated and is found to have the form 2 2   2 4GN M 15 GN M π S b ~GN M θ ' + 2 + 8c + 9 − 48 log 3 + ... (10) b 4 b 2b0 πb with cS being a constant which depends on the spin of the scattered particle. It is found that cS = 371/120, 113/120, − 29/8 for a massless scalar, photon or graviton, respectively. Here b is the impact parameter and log b0 is an infrared logarithm which we need not be concerned with here. This result has been calculated in the eikonal limit. This involves large impact parameter, and is the limit where the effective field theory is most applicable. Of course, quantum scattering leads to a range of bending angles and this bending angle formula has been defined by using the maximum of the eikonal phase. There are several implications which can be immediately seen. This phenomenon cannot be described by motion as different massless species respond differently. It is then not equivalent to a quantum modification to the metric. The trajectory of massless particles is used in flat space to define the lightcone. That cannot be done in this background again due to the species dependence. Moreover, in the details of the calculation we can see that the quantum evolution samples the gravitational field over many points in space, not just along a local geodesic. In loop diagrams, the massless propagators are not localized in spacetime, but propagate over large distances. This then is sensitive to the gravitational field over these distances. When averaged over all directions, this converts the leading 1/b dependence into the 1/b2 classical correction and the 1/b3 quantum correction. The nonlocality is manifest in non-analytic terms in the amplitudes. These cannot be Taylor expanded in the momentum. Taylor expansions of analytic terms can be written as local operators, with the higher momentum pieces being equivalent to derivatives of delta functions. The non-analytic and non-universal behaviors appear even in the gravitational couplings. This can be seen from the effects of long distance graviton propagators, as calculated in [11, 12], with the amplitude,

X N 2 iMX ' (Mω) ~ h κ2 15 M 15  q2  buS  q2  × − + κ4 + κ4 log − κ4 log (11) q2 512 |q| ~ 512π2 M 2 ~ (8π)2 µ2 3  q2  Mω i  q2  i + κ4 log2 − κ4 log . ~ 128π2 µ2 8π q2 M 2

The points to notice here is that the logarithms, and squares of logarithms, signal the non-locality. In addition, the coefficient buS differs for different species, much like the coefficient cS in the formula for the bending angle given above. This was calculated using a unitarity-based method, and the fact that this occurs in the graviton cuts is another indication that it is really quantum gravity participating in this non-locality and non-universality. Overall, this non-locality implies that the motion is not purely a geodesic. 4

Of course the eikonal amplitude is derived in a limit of a weak field and large impact parameter. The actual quantum modification for motion aroung the Sun is tiny, and for all effective purposes we can continue to use lightcones in situations where the quantum graviy effects are small. However, the lessons learned here carry over to stronger and more complicated gravitational backgrounds. The propagation of massless particles in loops will always make the interaction nonlocal - sampling the background field over a range of positions. The trajectory (however it is defined) will not be described by the classical geodesic. Like the eikonal amplitude, we should expect that the effects are different for different species. This rules out the possibility of defining an effective geodesic. Moreover the wave nature of the massless fields becomes more important at strong coupling and it is not possible to define an effective “bending angle” as was done in the simpler eikonal approximation. These simple considerations tell us that many of our familiar classical concepts do not carry over to quantum gravitational physics. For causality, it implies that we lose our standard definitions of causal behavior, even if the asymptotic observers are in regions which are flat enough to apply Minkowski definitions locally 3.

DEPARTURE FROM LOW ENERGY - TIME DELAY IN PHASE SHIFTS

One might be concerned that causality uncertainties from high energy effects could be manifest at low energy and spoil the causal properties of the effective field theory. One low energy manifestation is in the signs of coefficients of the local terms in an effective Lagrangian. This was first noted in non-gravitational effective Lagrangians [13]. Some of the higher order coefficients must be positive (in a given convention) in order to emerge from a causal UV complete theory. There has been an effort to extend these constraints to gravitational theories also. Here the arguments are not as clear as they invoke classical gravitational concepts such a lightcones, and so could be viewed with caution. However, we will show below that as one departs from low energy there are effects which also traditionally signal causality violation and which do depend on the sign of the coefficients, without directly invoking the lightcone. If there is a coefficient with the “wrong” sign in the low energy effective Lagrangian, does this upset causality in the effective field theory in the region where the effective field theory is valid? A recent analysis [14] shows that it does not. The point is that in the low energy limit one deals with fields with large wavelengths. The higher order operators in the theory are small perturbations in the effective field theory limit. While there can be time advances and superluminality generated by these operators, the effects are small enough to be unresolvable when working with large wavelength fields which are appropriate for the effective field theory. However, as one proceeds to higher energy, the effects of the higher order operators will become more apparent. Their effects can become relevant as one approaches the limits to the validity of the effective field theory, even if new degrees of freedom are not yet dynamically active. This can be seen in standard effective field theories, such as chiral perturbation theory. For example in the J = 0 channel of ππ scattering, the amplitudes approach the unitarity bounds well below the GeV scales where the quarks and gluons become active. The theory still respects unitarity in this limit [15], but becomes strongly coupled. If the chiral Lagrangian had “wrong” sign coeficients (it does not, because QCD is causal) they could show up in scattering phases in this transition region. We present an example from quadratic gravity which has this behavior. In quadratic gravity the sign of the coefficient of the Weyl squared term is uniquely fixed by the requirement that the massive pole is not a tachyon. The two possible signs correspond to tachyonic excitation or a massive unstable ghost (the Merlin resonance). We know of no way to make the tachyonic branch into a physically viable theory. However the unstable ghost branch leads to a unitary theory, albeit one with causality violation. We can see how this choice of sign influences physical observables by considering scattering of regular particles in the spin-two channel in which case the s-channel scattering goes through the Merlin resonance. The causality violation by the Merlin mode both dictates the sign of the higher order operator and also leads to a negative Wigner time advance in the scattering amplitude. In potential scattering, Eisenbud and Wigner derived an implication of causality in the scattering phase δ from a potential which vanishes identically outside of a radius R and is bounded ∂δ 2mR δT = 2 ≥ − . (12) ∂E p

3 We remark that the discussion of lightcone fluctuations here differ from those developed by Ford and collaborators, see for instance Refs. [27–29]. In such analyses, even though one can verify “faster than light” signals, there is no causality uncertainty since the system is no longer Lorentz invariant – the graviton state specifies a preferred frame of reference. 5

This can be found heuristically by Taylor expanding the outgoing wave phase shift

ei(kr−Et)e2iδ(E) (13)

−iE(t−2 ∂δ ) in powers of the energy and identifying the shift e ∂E as a time delay. The assumption that the potential vanishes outside of a radius R is needed because the incoming wave can start scattering before reaching the center of the potential. For relativistic gravitational scattering the potential formally has infinite range, but handwaving estimates based on the uncertainty principle can be made that the right hand side of this equation should be no more than the order of the inverse of the center-of-mass energy. In practice applications are generally made by neglecting the right hand side of the equation, such that the time delay should be positive. Explicit calculations with realistic causal interactions support this. This is the reason that the phase shift advances in a counter-clockwise fashion on the Argand diagram. However in gravitation scattering in the spin two channel, the phase motion is in the opposite direction. In the large N limit the spin two scattering proceeds through the direct s-channel graviton propagator4. This was originally analysed by Han and Willenbrock [17] independent of any UV completion - see also [15]. We have explored the reaction in more detail in the context of quadratic gravity. The basic point is that unitarity is maintained by the imaginary parts which arise in the graviton vacuum polarization diagram. However, those imaginary parts lead the phase to proceed clockwise on the Argand diagram. Within quadratic gravity, this motion is completed by a full resonance - the Merlin resonance - with clockwise motion of the phase past −90o. As detailed in the previous papers, one finds that the partial wave amplitude T2 has the form:

N κ2s T (s) = − eff D¯(s). (14) 2 640π where  κ2s κ2sN  s  iκ2sN  D¯ −1(s) = 1 − − eff ln + eff (15) 2ξ2(µ) 640π2 µ2 640π

2 Elastic unitarity requires ImT2 = |T2| . This is satisfied for an amplitude of the form

A(s) A(s)[f(s) + iA(s)] T (s) = = (16) 2 f(s) − iA(s) f 2(s) + A2(s) for any real functions f(s),A(s). Since the imaginary part in the denominator comes from the logarithm, the unitarity condition is a relation between the tree-level scattering amplitude which determines the A(s) in the numerator and the logarithm in the vacuum polarization which determines the imaginary part in the denominator. This is satisfied with N κ2s A(s) = − eff . (17) 640π There is a correlation between the unusual sign of the imaginary part in the propagator and the sign of the scattering amplitude. This combination allows unitarity to be satisfied. The phase motion in this case is clockwise. This is seen in the Argand diagram of Figure 1, which has been continued through the resonance. In practice, the phase remains very small until one approaches close to the resonance. This low energy portion of this phase is universal. One can see this by considering the amplitude itself, for which the absolute value is shown in Figure 2 for different values of the parameter ξ. One sees that the amplitude is very small in the universal region - note the logarithmic scale. The phase motion happens when near the narrow resonance and this corresponds to the UV completion of the amplitude in quadratic gravity. For small values of ξ, this UV completion occurs below the Planck scale. However, for larger values of ξ the amplitude can grow large before the UV completion happens. Of course, in other theories of gravity, the behaviour will resolve in different ways. The main point here is that this phase behavior corresponds to a time advance rather than time delay. When this goes through a resonance, this is the defining characteristic of the Merlin mode, which propagates backwards in time.

4 The invocation of the large N limit is useful because there is also a t-channel scattering. However the s-channel behaves as described here even away from the large N limit. 6

FIG. 1: Argand diagram for the phase shift in the J=2 channel.

FIG. 2: Absolute value of the scattering amplitude in the J=2 channel. The curves from bottom to top correspond to ξ2 = 0.1, 1.0, 10.

FLUCTUATIONS IN THE CAUSAL STRUCTURE

Now we would like to get back to the discussion regarding the signs of coefficients of local terms in an effective Lagrangian. This amounts to considering positivity constraints and effective lightcones [13]. We will discuss the consequences for the propagation of massless particles. For simplicity, assume that the gravitons are coupled with a massless scalar field π with leading interaction κ − h T µν (18) 2 µν 7 where T µν is the energy-momentum tensor associated with the scalar field: 1 T = ∂ π∂ π − η ∂ π∂κπ. (19) µν µ ν 2 µν κ By integrating out the gravitons, one finds the following contribution to the effective action

1 Z κ2 Z Z S = d4x ∂ π(x)∂µπ(x) − d4x d4x0 T µν (x)D (x − x0)T αβ(x0) (20) eff 2 µ 8 µναβ

0 where Dµναβ(x ) is the propagator associated with the field hµν . Let us discuss the form for this propagator in the framework of quadratic gravity. Using a suitable generalized de Donder-like gauge, and employing a Lehmann representation, the propagator can be brought to the form [34]

4 1  Z d q 0 1  D (x − x0) = η η + η η − η η e−iq·(x−x )D (q) = η η + η η − η η D (x − x0) µναβ 2 µα νβ µβ να µν αβ (2π)4 2 2 µα νβ µβ να µν αβ 2 Z ∞ Z ∞ 2 2 1 σ(s) 1 ρe(s) D2(q) = G(q ) + GeMerlin(q ) = ds 2 − ds 2 (21) π −∞ q − s + i π 0 q − s − i where σ(s) and ρe(s) are associated spectral functions. The second term arises in quadratic gravity due to the presence of higher-derivative terms – the source for the appearance of the Merlin modes. This choice is motivated by simplicity; indeed, the spin-2 piece of the propagator obtained from the Dµναβ has a similar form as the one considered in Ref. [30] 5. One can also envisaged such a propagator as the “double-copy” version of the higher-derivative Yang- Mills propagator considered in Ref. [31], after resorting to a Lehmann representation. We can rewrite the effective action as  2  Z 1 κ Z  0  S = d4x ∂ π(x)∂µπ(x) − d4x0 D (x − x0) 2∂µπ(x)∂ν π(x)∂0 π(x0)∂0 π(x0) − ∂ π(x)∂λπ(x)∂0 π(x0)∂ κπ(x0) . eff 2 µ 16 2 µ ν λ κ

The equations of motion read

2 κ Z  0  ∂µ∂ π − ∂µ d4x0 D (x − x0) 2∂ν π(x)∂0 π(x0)∂0 π(x0) − ∂ π(x)∂0 π(x0)∂ ν π(x0) = 0. (22) µ 4 2 µ ν µ ν

Let us expand our model around the solution ∂µπ0 = Cµ, where Cµ is a constant vector. The linearized equations of motion for small fluctuations ϕ = π − π0 around this background is given by

 2 Z ∞ Z ∞   µν κ σ(s) ρe(s) µ ν η + ds − ds C C + ··· ∂µ∂ν ϕ = 0 (23) 2π −∞ s 0 s

2 ν 1/2 where we kept only the leading contribution linear in ϕ and we considered that κ (C Cν )  1 so that all such terms ν can be neglected (in any case, the proper inclusion of such terms proportional to C Cν wil not change qualitatively our conclusions). This is the same linearized equation obtained from the following effective theory [13] 1 c L = ∂µπ∂ π + 3 ∂µπ∂ π∂ν π∂ π + ··· (24) eff 2 µ Λ4 µ ν whose equations of motion are given by 4c ∂µ∂ π + 3 ∂µ∂ π∂ν π∂ π = 0. (25) µ Λ4 µ ν Upon the same linearization procedure, one finds

 8c  ηµν + 3 CµCν + ··· ∂ ∂ ϕ = 0 (26) Λ4 µ ν

5 Actually, this is not the full form for the graviton propagator – there are also numerator terms containing projectors with explicit qµ dependence. But we disregard such terms since they cancel out in connected correlators. 8 which implies the following identification 2 Z ∞ Z ∞  c3 κ σ(s) ρe(s) 4 ≡ ds − ds . (27) Λ 16π −∞ s 0 s An expansion in plane waves yields 8c pµp + 3 (C · p)2 = 0. (28) µ Λ4

As discussed in Ref. [13], the absence of superluminal excitations requires that c3 > 0. Now, assume the narrow- width approximation. In the low-energy regime and away from the resonance, the first term in the hµν -propagator dominates and therefore the first term on the right-hand side of Eq. (27) is the most relevant in this situation. Hence the positivity constraint is satisfied. However, close to the Merlin resonance, it is the second term that dominates and as a result the positivity constraint is violated (spectral functions are taken to be positive by unitarity). So the scalar fluctuations experience a strong effective causal uncertainty near the Merlin resonance. A similar analysis can be carried out for the graviton field [35]. The linearized equation for the fluctuations ϕ suggests a natural effective “metric” (at least in the geometric optics limit) 2 Z ∞ Z ∞  8c3 κ σ(s) ρe(s) Gµν = ηµν − 4 CµCν + ··· = ηµν − ds − ds CµCν + ··· (29) Λ 2π −∞ s 0 s with which one could define effective lightcones and time evolution. This allows us to discuss standard causality conditions encoded in the following theorem, which we reproduce here [36]:

Theorem: A spacetime M is stably causal if and only if there exists a differentiable function f on M such that ∇µf = tµ is a past directed timelike vector field.

Roughly this theorem establishes that stable causality is equivalent to the existence of a global time function µ µ t = −δ0 on the spacetime. Since 2 Z ∞ Z ∞  κ σ(s) ρe(s) 0 2 G00 = 1 − ds − ds (C ) (30) 2π −∞ s 0 s we see that, in the low-energy regime tµ is is globally defined, non-degenerate and timelike. In particular, future timelike curves x(τ) satisfies dxµ dxµ dxν tν G < 0, G > 0 (31) dτ µν dτ dτ µν as expected. Moreover, close to the Merlin resonance, the second term in the hµν -propagator dominates and hence the second term in the expression for G00 is the relevant one. This implies that G00 > 1. Lightcones for the effective metric are strictly larger then those of ηµν . So now the question would be if f is well defined. Stably causal spacetimes do not admit closed timelike curves. In the presence of such an exotic objects, the time coordinate is not globally defined. In order to discuss this condition, it is best to consider the effective metric in spherical coordinates. The associated expression can be derived by using the usual tensor transformation rule ∂xk ∂xl G0 = G (32) ij kl ∂x0i ∂x0j where the primes denote the usual spherical coordinates, x01 = r, x02 = θ, x03 = φ, with x1 = x01 sin(x02) cos(x03) x2 = x01 sin(x02) sin(x03) x3 = x01 cos(x02). (33)

The metric component Gφφ is given by ∂x1 2 ∂x2 2 ∂x1 ∂x2 G = G + G + 2G φφ 11 ∂φ 22 ∂φ 12 ∂φ ∂φ  κ2 Z ∞ σ(s) Z ∞ ρ(s)   = − 1 + ds − ds e (C1)2 sin2 φ + (C2)2 cos2 φ − C1C2 sin 2φ r2 sin2 θ 2π −∞ s 0 s (34) 9

µ µ with φ = 0 and φ = 2π being identified as usual; the integral curves of the vector field η = δφ are closed. If we take C2 = 0, we find that

 2 Z ∞ Z ∞   κ σ(s) ρe(s) 1 2 2 2 2 Gφφ = − 1 + ds − ds (C ) sin φ r sin θ. (35) 2π −∞ s 0 s

In the low-energy regime, we see that Gφφ < 0. However, close to the Merlin resonance, Gφφ can change sign and hence φ can become a timelike coordinate. This signals the presence of closed timelike curves: ηµ becomes a timelike vector field. The violation of stable causality can also be seen directly in the full metric gµν = ηµν + hµν . If we choose to use the retarded Green’s function (whose spin-2 part will be presented in the next section) as a classical Green’s function giving the response to the external source given by the energy-momentum tensor Tµν , one finds that 1 Z g (x) = η − d4x0Dret (x − x0)T αβ(x0) µν µν 8π µναβ 2 4 2 κ   Z d q 0 κ   Dret (x) = η η + η η − η η e−iq·(x−x )Dret(q) = η η + η η − η η Dret(x) µναβ 2 µα νβ µβ να µν αβ (2π)4 2 µα νβ µβ να µν αβ 1 1 Dret(q) = − . (36) 0 2 2 2 2 2 0 0  (q + i) − q q − mr − iγθ(q ) θ(q ) − θ(−q ) The second term corresponds to the propagation of the effect backward in time. If we consider a conformally invariant external source, one finds that

κ2 Z g (x) = 1 − d4x0Dret(x − x0)T (x0). (37) 00 8π 00

If matter obeys the weak energy condition, then the second term of Dret(x − x0) can produce a broader lightcone. Following a similar analysis as above, one can show that closed timelike curves can be generated by this geometry due to the presence of the Merlin resonance. Hence stable causality is violated and causal uncertainty takes place near the Planck scale. In summary, the effect of the Merlin modes is to induce strong quantum fluctuations of the background lightcones near the Planck scale. Such fluctuations also open out the cones, therefore allowing for the construction of closed timelike curves. This implies that stable causality is violated near the Planck scale and as a result one verifies some causal uncertainty at high energies. Once again we see that the lightcone is not a well-defined concept when considering quantum-gravity effects.

CAUSAL UNCERTAINTY IN QUADRATIC GRAVITY

We now study with more detail the emergence of causality uncertainties in the specific context of quadratic gravity. As well known, in a causality preserving quantum field theory, the Pauli-Jordan function has support on the lightcone. This is usually refer to as the requirement of microcausality: Any two local observables at spacelike separation must commute [2]. We will see how this can be violated in quadratic gravity. Let us begin our study by considering the retarded and advanced Green’s functions. Usually such functions vanish for spacelike separations. Taking into account the one-loop vacuum polarization, the spin-2 part of the retarded Green’s function is given by [30, 32]

Z 4 ret 0 2 (2) d q −iq·(x−x0) ret iD (x − x ) = κ Pe (∂x) e iD (q) µναβ µναβ (2π)4 2 1 1 Dret(q) = − 2 0 2 2 2 2 2 0 0  (q + i) − q q − mr − iγθ(q ) θ(q ) − θ(−q ) 2 2 2 2  2  2 2 2 2ξ (mr) ξ (mr)Neff |q | 2ξ (mr) mr = 2 + 2 2 ln 2 ≈ 2 κ 320π κ mr κ ξ2(m2)N γ ≈ m2 r eff (38) r 320π 10 where we consider that, in the weak-coupling limit ξ << 1, γ is a small number so that one may drop the term 2 2 (ξ Neff/320π) and the logarithmic term. Contour integration will lead us to the result ( Z 3  −i(ωq t−q·x) i(ωq t−q·x)  ret 2 (2) d q e e Dµναβ(x) = κ Peµναβ(∂x) 3 −iθ(t) − (2π) 2ωq 2ωq   −i(E t−q·x) i(E t−q·x) ) e q e q − γ|t| + iθ(−t) − e 2Eq (39)   γ   γ  2 Eq + i 2 Eq − i 2Eq 2Eq

p 2 2 where ωq = |q| and Eq = q + mr . So we see that not only there is an unusual term for t < 0 but also such a term fails to vanish for spacelike separations (the easiest way to see this is to consider a frame in which the separation is purely spatial, that is, t = 0). In turn, for the advanced Green’s function, a similar reasoning lead us to the following result: ( Z 3  −i(ωq t−q·x) i(ωq t−q·x)  adv 2 (2) d q e e Dµναβ(x) = κ Peµναβ(∂x) 3 iθ(−t) − (2π) 2ωq 2ωq   −i(E t−q·x) i(E t−q·x) ) e q e q − γt − iθ(t) − e 2Eq . (40)   γ   γ  2 Eq − i 2 Eq + i 2Eq 2Eq Hence one verifies the presence of an unusual term for t > 0. In addition, the advanced Green’s function also fails to vanish for spacelike separations. 0 To carefully implement a study on causality, one must duly calculate the Pauli-Jordan function Gµναβ(x, x ) = 0 [hµν (x), hαβ(x )]. Therefore, let us consider the associated Wightman functions. Again we investigate the simple case of a massless scalar field coupled with gravity. The full gravitational Pauli-Jordan function is given by 0 + 0 − 0 Gµναβ(x, x ) = Gµναβ(x, x ) − Gµναβ(x, x ) (41) ± where Gµναβ are Wightman functions. From the results discussed at length in the Appendix, we find that Z 0 0 0 00 0ret 00 γδ 00 0 Gµναβ(x, x ) = Gµναβ(x − x ) + dx Dγδαβ(x − x )F µν (x , x ) (42)

0 0 where Gµναβ(x − x ) is the unperturbed Pauli-Jordan function and we have defined

 δL−   δL+  F γδ (x00, x0) = int [Φ−(x0)] h+ (x00) + int [Φ+(x00)] h− (x0) . (43) µν δh− (x0) µν δh+ (x00) µν γδ h−=h+=h γδ h−=h+=h 0ret 0 These terms will be contribute to S-matrix elements. In this equation, Dγδαβ(x − x ) is the unperturbed retarded Green’s function and Lint describes gravitational interactions. We use the in-in formalism, so that fields and sources are doubled (which clarifies the ± appearing above) and Φ± denotes the collection of the fields. For a thorough discussion of terminology and notations, see the Appendix. As we have shown above, the retarded Green’s function, in addition to the usual term, contains an anomalous 0ret term associated with propagation backward in time. Since Dγδαβ is an unperturbed Green’s function, the spin-2 contribution must have the form " ret 0 2 (2) 1  0 0  1  0 0  D (x − x ) = κ Pe (∂x) − δ t − (t − |x − x |) + δ t − (t + |x − x |) µναβ µναβ 4π|x − x0| 4π|x − x0|  √ # m0J1 m0 σ0 0 − θ(t − t )θ(σ0) √ (44) 4π σ0

2 2 2 where m0 = 2ξ /κ . For our purposes the term with the Bessel function J1 will not be important so we will not study its contribution in detail. The Pauli-Jordan function reads 2 Z 3 00 " 0 0 0 κ (2) d x γδ 00 0 00 0 Gµναβ(x, x ) = Gµναβ(x − x ) + Peγδαβ(∂x) 00 −F µν (t − t ; x − x ) 4π |x − x | t00=t−|x−x00| # γδ 00 0 00 0 + F µν (t − t ; x − x ) + ··· (45) t00=t+|x−x00| 11

0 where the ellipsis correspond to the contribution coming from the Bessel function J1. Now consider that x = −x = rˆn. Furthermore, consider the Pauli-Jordan function for large values of r. Then one has that

|x − x00| ≈ r − x00 · ˆn

Introducing the Fourier transform

Z d4p F γδ (x) = F γδ (p)e−ip·x µν (2π)4 µν one obtains 2 Z 3 00 Z 4 0 0 0 κ (2) d x d p γδ h −ip0(t−t0−r+x00·ˆn) ip·(x00−x0) Gµναβ(x, x ) ≈ G (x − x ) + Pe (∂x) F (p) −e e µναβ 2π γδαβ |x − x0| (2π)4 µν 0 0 00 00 0 i + e−ip (t−t +r−x ·ˆn)eip·(x −x ) + ··· 2  Z 0  0 0 κ (2) 1 dp h γδ 0 0 ip0z γδ 0 0 −ip0z0 i = G (x − x ) − Pe (∂x) F (p , p ˆn) e − F (p , −p ˆn) e + ··· µναβ 2π γδαβ |x − x0| 2π µν µν (46) where z = |x−x0|−(t−t0) and z0 = |x−x0|+(t−t0). Let us assume that t−t0 > 0 and z > 0. Then the first term is the most interesting one. One may evaluate the first term using contour-integration methods. Consider an appropriate contour above the real axis. If the integral gives a zero value, that means that the integrand is an analytic function in the upper half p0 plane. As a consequence, the Pauli-Jordan function will vanish for a spacelike separation z > 0. γδ 0 However, F µν (p) is in general not an analytic function in the upper half p plane due to the contribution coming from the Merlin modes. Hence the full Pauli-Jordan function fails to vanish for spacelike separations in quadratic gravity.

SUMMARY

Because the size of quantum gravitational effects grows with energy, metric fluctuations on short scales are expected to be large, if the metric even survives as a dynamical variable. On these scales, standard ideas of causality likely no longer make sense. While we cannot treat the deep quantum regime with reliability, we can see some causality uncertainty arising at lower energy where the calculations are more meaningful. We have shown here through the discussion of many examples that causal uncertainty in field theory should be expected in quantum gravity. This emerges even at low energies, as the analysis of the effective field theory of gravity clearly shows that the concept of a lightcone becomes problematic. We have also discussed this issue in the context of quadratic gravity, where two possible causal flows are present at energies close to the Planck scale. In this case it is clear how causality in a traditional sense should be envisaged as an emergent macroscopic phenomenon. One traditional way of thinking on causal uncertainty concerns the analyticity of the S-matrix. Namely, when amplitudes fail to satisfy the standard analyticity axioms of S-matrix theory, one verifies some sort of causality violation. This is exactly what happens in Lee-Wick-like theories, where the poles associated with Merlin modes are located on the physical sheet. Here we have shown how this impacts the calculation of the Pauli-Jordan function, which embodies the concept of microcausality in field theory. In our evaluation, we have demonstrated that the full Pauli-Jordan function fails to vanish for spacelike separations in quadratic gravity. One other possible way to study causal uncertainty is through an analysis of the Shapiro’s time delay. Indeed, higher derivative corrections to the graviton three-point function lead to a potential causality violation unless we get contributions from an infinite tower of extra massive particles with higher spins, which implies a Regge behavior for the amplitudes – this is verified in a weakly coupled [37, 38]. This also happens in large N QCD coupled to gravity [39]; in this case there is also an alternative (but similar) resolution: The higher-spin states well below the Planck scale can come from the QCD sector. All such examples displaying causality violations (and the associated resolutions) come from higher-derivative corrections to gravitational interactions. The situation in quadratic gravity is somewhat different; if, on one hand, 3-particle amplitudes involving only physical gravitons do not display contributions coming from higher-order derivative terms (and this result generalizes to an arbitrary number of external gravitons) [31, 40], on the other hand the existence of causal uncertainty comes from the fact that there are two sets of modes that follow different causal directions; that is, if one takes such sets in separation, there should 12 be no issues with causality [3, 4]. The problem lies when one takes them together – “dueling arrows of causality” near the Planck scale. All associated amplitudes will present normal behavior, which is reflected by the fact that the theory is unitary [10]. Moreover, we expect Shapiro’s time delay to exhibit such features. So a potential “time advance” instead of time delay can be verified when one takes a theory containing normal modes together with Merlin modes. The full analysis of the Shapiro’s time delay in quadratic gravity would be interesting to explore, and we hope to do this in the near future. The uncertainty in standard ideas of casuality can also be seen in gravitational models where there is a causal order, such as causal sets [41]. The reasoning is similar in that there are quantum fluctuations in the elements which blurs the ordering on the average. It would be interesting to see if similar uncertainties could be quantified in highly quantum numerical simulations in schemes such as Causal Dynamical Triangulations [42]. Many discussions of quantum gravity continue to use classical concepts such as lightcones and background metrics. These may be permissable in regimes where the quantum effects are tiny. But we are often interested in regimes where quantum effects are important, and then these classical ideas are likely illegitimate. The causality uncertainty which we have explored is one of the newer quantum features which arises. We need to develop ways of understanding the deep quantum regime which acknowledges the changes which occur in some of our standard expectations.

Acknowledgements

The work of JFD has been partially supported by the US National Science Foundation under grant NSF-PHY18- 20675. The work of GM has been partially supported by Conselho Nacional de Desenvolvimento Cient´ıficoe Tec- nol´ogico- CNPq under grant 307578/2015-1 (GM) and Funda¸c˜aoCarlos Chagas Filho de Amparo `aPesquisa do Estado do Rio de Janeiro - FAPERJ under grant E-26/202.725/2018.

Appendix: Derivation of the Wightman functions in the in-in formalism

In this Appendix we discuss in detail the Wightman functions in the in-in formalism. We also shall derive the necessary Schwinger-Dyson equations. For simplicity, consider a coupling with a massless scalar field:

Z  1 1  S = d4x − hµν 2 hαβ − φ2φ + C¯ M α Cβ + L [h, φ, C, C¯] (47) 2 µναβ 2 α β int where as above 2µναβ is a shorthand for the graviton kinetic terms (with the gauge-fixing contributions already included). In the above equation, Cβ is the standard Faddeev-Popov ghost field. After doubling the fields and sources, one obtains

Z ( Z ± ± ± µν 1/2 ∗ 4  µν + + + + α ¯α + Z[J ,K , σ±, σ¯ ] = (det D ) DΦ+DΦ− exp iS[Φ+] − iS [Φ−] + i d x J+ hµν + K φ +σ ¯α C+ + C+σα

Z ) 4  µν − − − − α ¯α − − i d x J− hµν + K φ +σ ¯α C− + C−σα (48)

± ± α ¯α ∗ where Φ± denotes the collection of the fields hµν , φ ,C±, C± and S indicates that in this functional i → −i. Gravitational Green’s functions can be derived in the usual way, namely

m n 1 1 1 δ δ ± ± ± − Z[J ,K , σ±, σ¯ ] m+1 n+1 µ1ν1 µmνm α1β1 αnβn Z[0] (−i) (i) δJ (y ) ··· δJ (y ) ± ± ± − 1 − m δJ+ (x1) ··· δJ+ (xn) J =K =σ±=¯σ =0 D n o n oE = T¯ h− (y ) ··· h− (y ) T h+ (x ) ··· h+ (x ) (49) µ1ν1 1 µmνm m α1β1 1 αnβn n where T¯ denotes anti-time-ordering operation. In particular, for n = m = 1 one finds the Wightman function

1 δ δ ± ± ± − 0 + + 0 − 0 − µν 0 αβ Z[J ,K , σ±, σ¯ ] = hhµν (x )hαβ(x)i = Gµναβ(x , x) = Gµναβ(x, x ). Z[0] δJ (x ) ± ± ± − δJ+ (x) J =K =σ±=¯σ =0 (50) 13

Likewise, by reversing the order of the functional derivatives one gets

1 δ δ ± ± ± + − 0 + 0 − µν αβ Z[J ,K , σ±, σ¯ ] = hhµν (x)hαβ(x )i = Gµναβ(x, x ). (51) Z[0] δJ (x) 0 ± ± ± + δJ− (x ) J =K =σ±=¯σ =0 Now let us derive the Schwinger-Dyson equation for the gravitational Wightman functions. We start with the Schwinger-Dyson differential equations, given by     δS± δ δ δ δ ± ± ± ± µν −i ± , −i ± , −i ± , i ± + Jµν Z[J ,K , σ±, σ¯ ] = 0 (52) δh± δJ δK δσ¯ δσ where S± = S[Φ±]. Performing the functional derivative, one finds that

( ±   ) δ δLint δ δ δ δ ± ± ± ± i2µναβ ± + µν −i ± , −i ± , −i ± , i ± + Jµν Z[J ,K , σ±, σ¯ ] = 0. (53) δJαβ δh± δJ δK δσ¯ δσ

Performing a further derivative with respect to δ/δJ∓, one finds that

δ2Z δL±  δ δ δ δ  δZ i2x + int −i , −i , −i , i = 0. (54) µναβ ∓ 0 ± µν ± ± ± ± ∓ 0 δJγδ(x )δJαβ(x) δh± (x) δJ δK δσ¯ δσ δJγδ(x ) By taking further derivatives with respect to the external sources one can obtain the complete set of Schwinger-Dyson equations. In terms of correlation functions, one finds (setting all external sources to zero):

 δL+  x γδαβ 0 int + γδ 0 −i2µναβG− (x, x ) − i µν [Φ (x)] h− (x ) = 0 δh+ (x) h−=h+=h  −  0 δL −i2x Gγδαβ(x, x0) + i int [Φ−(x0)] hγδ(x) = 0. (55) µναβ + µν 0 + δh− (x ) h−=h+=h Given appropriate boundary conditions, the solutions of the above equations read

Z  δL+  G− (x, x0) = G0− (x − x0) − dx00D0ret (x − x00) int [Φ+(x00)] h− (x0) µναβ µναβ γδαβ δh+ (x00) µν γδ h−=h+=h

Z  δL−  G+ (x, x0) = G0+ (x − x0) + dx00D0ret (x − x00) int [Φ−(x0)] h+ (x00) (56) µναβ µναβ γδαβ δh− (x0) µν γδ h−=h+=h

0± 0 where Gµναβ(x − x ) obeys the equation

µναβ 0± 0 2x Gαβγδ(x − x ) = 0 (57)

0ret 0 and Dγδαβ(x − x ) is the unperturbed retarded Green’s function which obeys

µναβ 0ret 0 µν 0 2x Dαβγδ(x − x ) = I γδδ(x − x ) (58) where (in a flat background) 1  I ≡ η η + η η . (59) µναβ 2 µα νβ µβ να In possession of Wightman’s functions, one can explicitly construct an expression for the Pauli-Jordan function, given by Eq. (41).

∗ Electronic address: [email protected] † Electronic address: [email protected] 14

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