Modular Hamiltonians for deformed half-spaces and the averaged null energy condition

Journal of High Energy Physics, Sep 2016

We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on \( {\mathrm{\mathbb{R}}}^{1,d-1} \). We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.

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Modular Hamiltonians for deformed half-spaces and the averaged null energy condition

HJE Modular Hamiltonians for deformed half-spaces and the averaged null energy condition Thomas Faulkner 0 1 2 Robert G. Leigh 0 1 2 Onkar Parrikar 0 1 2 Huajia Wang 0 1 2 0 Department of Physics, University of Illinois , USA 1 the second term comes from @R 2 To see this, we focus on a particular term @ We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum eld theories on R1;d 1. We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at rst order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories. AdS-CFT Correspondence; Field Theories in Higher Dimensions - 1.1 Setup & summary of results Modular Hamiltonian for deformed half-space Reduced density matrix Modular Hamiltonian 1 Introduction 2 3 4 5 2.1 2.2 3.1 3.2 5.1 5.2 Averaged null energy condition Positivity of KbA0 Computing hKbA0 i KbA hKbAi Modular Hamiltonians in AdS/CFT Discussion Sharpening the argument Generalizations A Cuto at the entangling surface A is the reduced density matrix of the state over the subregion A. The modular Hamiltonian is, in general, a complicated, non-local operator and not of much practical use. However, the situation greatly simpli es for the vacuum state in the case of certain special symmetric subregions. For instance, the modular Hamiltonian for a half-space in relativistic quantum eld theories takes a very simple form; it is the restriction to the half space of the generator of boosts which preserve the entangling surface [1], and consequently generates a local and geometric modular ow. A similar construction is also possible for spherical subregions in conformal eld theories [2], for null slabs in the case where the vacuum state is de ned with respect to the generator of null translations on a null hypersurface [3, 4] etc. Recently, it has been argued that the modular Hamiltonian for states with classical was made in [7] (following previous work in [8{12]), where perturbative techniques were used to study the shape-dependence of entanglement entropy in conformal eld theories. In the present paper, we adapt these techniques to study modular Hamiltonians for deformed half-spaces in relativistic (not necessarily conformal) quantum eld theories. The study of shape dependence of entanglement is an important task for several reasons. The entanglement structure of quantum systems is highly constrained by powerful inequalities, such as strong subadditivity of entanglement entropy, positivity and monotonicity of relative entropy, etc. In many situations, these entanglement inequalities further imply fundamental constraints on the properties of quantum eld theories. For instance, the strong subadditivity property was used in [13, 14] to prove an entropic version of the c-theorem for renormalization group ows in two and three dimensions. Similarly, the properties of relative entropy have been used to prove several interesting results such as the Bekenstein bound [15], the generalized second law for causal horizons [16] and the covariant entropy bound in the context of semi-classical gravity [3, 4]. Entanglement inequalities have also been shown to constrain the bulk geometry in states with classical gravity duals [17{20]. The entanglement inequality which will be relevant for our purpose is that the full modular Hamiltonian for the vacuum state KbA = KA KAc (1.2) (i.e., the di erence between the modular Hamiltonian of the subregion A and that of the complementary subregion Ac) satis es a \monotonicity" property under inclusion [21]. This means that if we shrink the subregion A, then the corresponding change in the full modular Hamiltonian KbA is a negative semi-de nite operator. This property in fact follows from the monotonicity of relative entropy, as was shown in [22]. In the present work, we will show that this monotonicity property of the full modular Hamiltonian along with perturbative results on the shape dependence of the modular Hamiltonian allow us to prove another fundamental constraint, namely the averaged null energy condition (ANEC) Z 1 1 1However, there are alternative proposals for point-wise quantum energy conditions. See for example [27{29]. { 2 { for many special cases such as free scalar and Maxwell elds in general dimension [30{32], arbitrary quantum eld theories in d = 2 with a mass gap and some assumptions on the stress tensor [33], CFTs with classical gravitational duals in general dimension [34], etc. Wall has also argued that the ANEC holds true for free or superrenormalizable eld theories in general dimension [16]. In summary, there is substantial evidence so far to suggest that the ANEC is satis ed by generic quantum eld theories on Minkowski space-time, but a general proof has been missing hitherto (although see [35] for an argument involving certain assumptions on the OPE of non-local operators) | in this paper, we will partially ll this gap. On the other hand, the ANEC is known to be violated in general curved space-times, but an alternative proposal called the self-consistent achronal ANEC exists in this case | see [36{38] and references there-in for further discussion. While this is out of the scope of the present paper, our results can nevertheless be extended to prove the ANEC along static bifurcate Killing horizons even in curved space-times. There is another motivation for trying to prove the ANEC in Minkowski space-time. In [39], Hofman and Maldacena (HM) showed that in a conformal eld theory the validity of the ANEC in a certain class of states created by operator insertions implies bounds on the coe cients appearing in the three-point correlation functions of that CFT. For instance, in d = 4 they used this to derive a bound on the ratio of central charges 1 3 a c 31 18 (1.4) where a and c are the coe cients of the Euler density term and the Weyl tensor squared term in the conformal anomaly; tighter bounds can be obtained by imposing supersymmetry. While the assumption of the ANEC was considered reasonable, in the original paper no proof was given. Since then, there have been several attempts at a proof of the HM bounds with varying levels of success [35, 40, 41]. In particular, using analytic bootstrap methods the HM bounds were proven for a class of three-point functions in [42], building on the work of [43, 44]. These methods take as an input crossing symmetry and re ection positivity and apply these principles to various four point functions in a light-cone limit to delicately extract the HM bounds. In particular in this guise the HM bounds were related to causality properties of correlation functions in a shockwave background [44].2 In contrast, we will show that the general HM constraints on CFT three-point functions can be extracted directly from the three-point function itself - when the three-point function is interpreted as calculating some modular energy of the CFT in an excited state. Overall it is satisfying to see the ANEC, and consequently the Hofman-Maldacena bounds, arise as a natural consequence of the fundamental constraints satis ed by the entanglement structure of the vacuum. 1.1 Setup & summary of results We now outline the calculation we are interested in, and present a brief summary of our results. Consider the density matrix j ih j corresponding to a pure state de ned on the 2In theories with gravity duals, the HM bounds have also been shown to be related to bulk causality constraints [34, 35, 45, 46]. { 3 { product h = hA density matrix Cauchy surface . Let us partition into two subregions A and its complement Ac. For local quantum eld theories, we expect the Hilbert space h to factorize into the tensor hAc . If this is the case, we can trace over hAc to obtain the reduced A = TrAc (j ih j ) which contains all the relevant information pertaining to the subregion A. The entanglement entropy between A and Ac is de ned as the von Neumann entropy of A In this context, the boundary @A of A is referred to as the entangling surface. The modular Hamiltonian (also known as the entanglement Hamiltonian) KA is de ned as Similarly, we can also de ne the modular Hamiltonian corresponding to the region Ac, which we denote KAc . We can combine KA and KAc into another useful operator: (1.5) (1.6) (1.7) (1.8) (1.9) (1.10) (1.11) (1.12) which we will refer to as the full modular Hamiltonian. In this paper, we will primarily study the operators KA; KAc and KbA for the vacuum state of a relativistic quantum eld theory (as such we drop the label from now on), with the region A being a slightly deformed half-space. To specify the geometry in more detail, let us pick global coordinates x = (x0; x1; ; xd 1) = (x0; x) on R1;d 1, where x0 is the time coordinate, and x denotes spatial coordinates. Pick the Cauchy surface x0 = 0, and consider the half space A0 given by The vacuum modular Hamiltonian for the half space takes a particularly simple form [1] i.e., it is the generator of boosts which preserve the entangling surface restricted to the region A0, a result known as the Bisognano-Wichmann theorem [1]. Correspondingly, the full modular Hamiltonian is given by the full boost generator KbA0 = 2 Z d d 1x x1 T00(0; x) Note that KbA0 is a conserved charge, and as such annihilates the vacuum KbA0 j0i = 0. (This later property is true for more general regions as well.) One can consider a small deformation of the region A0 to KA = ln A KbA = KA 1Ac 1A KAc A0 = n x 2 R1;d 1jx0 = 0; x1 > 0 : o Z A0 KA0 = 2 d d 1x x1 T00(0; x) + constant SEE[ ; A] = TrA A ln A : A = n x 2 R1;d 1 x0 = 0; x1 > (~x)o { 4 { c c H+ Also shown are the Rindler horizons H directions ~x are implicit.) line), such that D(A) (darker shaded region) is contained inside D(A0) (lighter shaded region). corresponding to the regions A0 and Ac0. (The transverse where (~x) is a smooth function of the (d 2) transverse spatial coordinates (parametrizing the entangling surface), collectively denoted by ~x = (x2; ; xd 1). The deformation is special in that it is restricted within the original Cauchy surface. We can generalize this to also include time-like deformations (see gure 1) A = n x~ 2 R1;d 1 x~0 = 0(x1; ~x); x~1 = x1 + 1(x1; ~x); x1 > 0 : (1.13) o We identify the in nitesimal as the deformation vector eld and pick to point inward, i.e. D(A) D(A0) (where D denotes the domain of dependence). Our primary results in this paper are as follows: (i) We will rst show that the modular Hamiltonian KA, up to rst order in the shape deformation, is given by Z H+ Z H T + 2 Z A0 U KAU y = KA0 2 where U : hA ! hA0 is a unitary transformation the details of which we will specify later, H are the future and past Rindler horizons of D(A0) shown in gure 1, and appearing in the second and third terms above are components of the vector eld on the entangling surface in light-cone coordinates x = x0 x1. Similar expressions can also be written for KAc and KbA. (ii) We will then consider the expectation value h jKbAj i in states of the form (1.14) (1.15) j i = e H O (0; x)j0i { 5 { and linear combinations thereof, where H is the Hamiltonian and O is an arbitrary local operator whose quantum numbers (dimension, spin etc.) are collectively denoted by .3 The factor of e H is added to make these states normalizable. In a CFT this class of states is a basis for the entire Hilbert space, via the state-operator mapping. For a general QFT similar statements should hold. In fact, there is no obstruction to generalizing our argument to include states created by many local and even non-local operators inserted throughout the lower half Euclidean plane. Further, we could also insert the operators in real time. In the interest of simplifying our presentation we choose to represent our state via a single operator insertion on the Euclidean section, although we expect all our conclusions to go through even in the more general case. We then show that equation (1.14), along with the positivity of the operator KbA0 KbA (i.e. monotonicity under inclusion, which recall follows from the monotonicity of relative entropy) implies the averaged null energy condition (ANEC) Z 1 1 remarks below equation (1.15), our derivation of the ANEC also applies to these states. (iii) Finally, we also discuss the (vacuum) full modular Hamiltonian for deformed halfspaces in CFTs with classical gravity duals, which allows us to make contact with the recent proposal by Ja eris-Lewkowyzc-Maldacena-Suh (JLMS) [6] for the holographic dual to the modular Hamiltonian. At this point we should mention that in continuum quantum eld theory there are signi cant ultraviolet (UV) issues associated with the de nition of the reduced density matrix for a region, often resulting in divergences for entanglement entropy and modular energy which are local to the entangling surface. These issues and associated divergences are however not present for quantities like the relative entropy, and the full modular Hamiltonian [47, 48]. Since this is ultimately what we are interested in, and in the interest of simplicity of presentation, we will for the most part suppress the need for a UV cuto at the entangling surface. Indeed the answers we will nd will be nite, partly justifying this approach. For further discussion on how to include such a UV cuto in our calculation, see appendix A, where we will argue for the irrelevance of the details of such a cuto beyond 3In the case of tensor operators, we contract them with appropriate polarizations, for instance O (x) = its existence. 1 2 s In this section, we give an explicit formula for the modular Hamiltonian KA of the vacuum state over a deformed half-space, to rst order in the shape deformation. interest of generality, let us instead consider a more general state rather than the vacuum4 j i = X c j i = e H X c O (0; x)j0i; ( > 0) (2.1) HJEP09(216)38 This state can be constructed similarly as a sum over path-integrals, but with the operator O in the term proportional to c . The reduced density matrix below (x0E ! 0 ; x1 > 0) the region A0 (see gure 2) corresponding to j i, associated with the undeformed half-space A0 is constructed as a Euclidean path integral with speci ed eld con gurations above (x0E ! 0+; x1 > 0) and where we have collectively denoted all the elds integrated over in the path integral as , 1 and j 0i; 0i 2 hA0 are eigenstates of the eld operator restricted to A0. The prefactor NA0; is added to ensure the normalization of the density matrix, i.e. TrA0 A0; = 1. We have explicitly displayed the dependence of the reduced density matrix on the metric 5 through the path integral measure (which we assume is di eomorphism invariant), the action and the normalization. For convenience, we will henceforth use the notation (inside path-integrals) X = X and below A, and with a real-time fold around x0E = 0 in the case of time-like deformations ( 0 6= 0). We can deal with this path integral by performing a di eomorphism f : x ! x , which maps A to A0. We can take to be non-vanishing (corresponding to non0 trivial f ) only within a small region jxE j < ` (for some ` cuto ). Of course, such a di eomorphism has a non-trivial action on the background metric , but much larger than the 4Later, we will also need to compute the expectation value hKAi = TrA AKA; in the excited state j i5; Hsoerweebyderive the reduced density matrix section which is used in constructing the Euclidean path integral is . we are denoting the metric in real time. Of course the corresponding metric on the Euclidean A along the way while setting up the calculation for KA. g = (f 1 ) { 7 { (2.3) (2.4) x † x 6The unitarity follows from the di eomorphism invariance of the measure: (f ) [D ] [D (f 0)] = [D 0]g : at x0E = the state j i, over the original half space A0 (solid blue line). The operator insertions are marked . The black dot is the entangling surface (with transverse directions ~x implicit). where denotes the pullback. We claim that the reduced density matrix over A (with the metric ) is given by A; = U y A0;g U restricted to A by j i; j i where U is a unitary transformation, and undeformed half-space, but with the deformed metric g. A0;g is the reduced density matrix over the We now give a quick formal proof of this claim. If we denote the eigenstates of 2 hA, then we can construct a unitary 6 operator U : hA ! hA0 U = [D ] j(f 1 ) ih j Then the claim (2.5) can be checked explicitly by a series of manipulations on the pathintegral de nitions of the above density matrices [49] 1 1 Z Z h j A; j i = NA; [D ] X e S[ ; ] += = Z (f ~)+= (f ~) = = NA; = NA0;g ~ =(f 1) 1 Z ~+=(f 1) = h(f 1 ) j A0;gj(f 1 ) i = h jU y A0;gU j i: [D(f ~)] X e S[ ;(f ~)] [D ~]g X e S[g; ~] x0E = ⌧ x0E = ⌧ A; , the second equality is obtained by changing variables = f ~ inside the path integral, while the third equality follows from the assumption that the measure is di eomorphism invariant. We have throughout used the fact that the operator insertions (denoted by X, following the de nition (2.3)) are away from the region where the di eomorphism f has non-trivial support, and so f acts trivially on these operators. In the case where f is an in nitesimal di eomorphism, we can obtain a perturbative formula for A0;g. Writing the deformed metric on the Euclidean section as where is appropriately Wick rotated to Euclidean space, we obtain U A; U y = A0; + ddx g (x) A0; coordinate in the (x0E ; x1) plane, then where g ), and T is the angular-ordering operator: if 2 (0; 2 ) is the angular g = ) + O( 2 ) (2.8) where H is the Heaviside step function. For the special case j i = j0i, we then obtain space for the vacuum state We are now in a position to construct the modular Hamiltonian over the deformed halfKA; ln A; = U y (ln A0;g) U = U yKA0;g U In order to perturbatively expand the right hand side in powers of , we use the resolvent trick ln A0;g = 1 + which together with equation (2.11) gives KA0;g = KA0; + KA0 + O( 2 ) In the interest of simplifying notation, we will henceforth drop the explicit reference to the Minkowski metric on A0; , and simply refer to it as A0 . It is possible to perform { 9 { 0 xE x 1 Rb and the region outside is Re. Also shown is the brach-cut @Re . the integral by going to the spectral representation (for details see [7, 12], where similar calculations were performed). The result is ddx g (x) A0 is=2 : T : (x) iAs0=2 (2.17) Z 1 A0 Since the operator is=2 generates modular evolution in Rindler time s, we see that the stress tensor is e ectively liberated from the Euclidean section and inserted in real time. We now arti cially split the integration region over which the stress tensor is inserted into two parts: a small solid cylinder Rb of radius b around the entangling surface, and its complement Re. We will later show that the contribution from inside the cylindrical neighborhood vanishes in the limit b ! 0. The region of integration is thus R = Rb [ Re where we should remember that R contains a branch cut along the surface A0. We now write g and integrate by parts on the region Re KA0 = is=2 A0 Z R e : T : d + Z A0 is=2 + Kb (2.18) The rst term involves the divergence of the stress tensor; in the absence of other operator insertions in the region where has support, we can drop this term. (Indeed, expectation values in states of the form (1.15) which we will be interested in have precisely this property, since has no support at the location of the operators O .) The second term is integrated and gets two types of contributions: (i) from the boundary surface,7 and (ii) above and below the region A0, which we will refer to as the branch cut (see gure 3). Finally Kb represents the contribution (iii) from inside the cylinder Rb. 7Not to be confused with the UV cuto surface that we discuss in appendix A. (i) Imaginary cuto surface: let us rst deal with the term supported on the surface @Rb. It is convenient to switch to complex coordinates z = x 1 ix0E; z = (x1 + ix0E): + Tzz(xs)e i + e 2s+i Tzz(xs) z HJEP09(216)38 In these coordinates, we nd where, is=2 A0 n T (x) is=2 A0 = e2s i Tzz(xs) + Tzz(xs)ei z xs = (b sin( + is); b cos( + is); ~x) Further, n is the (inward pointing) unit normal to @Rb and z and z are the components of the vector eld close to the entangling surface in holomorphic coordinates We now proceed by shifting the s integration contour s ! s + i in order to remove the dependence from the stress tensor. We do this after switching the order of integration so that the s integral comes before the integral. This step assumes analyticity in the complex s plane and that the contributions from s ! which can be justi ed in a spectral representation of (2.18). This gives 1 vanish, (2.19) (2.20) (2.21) (2.22) (2.23) (e2sTzz : Tzz :) zei + (: Tzz : e 2sTzz) ze i (2.24) where now these stress-tensors are evaluated at x0E = ib sinh(s); x1 = b cosh(s). We can now perform the integral using Z 2 0 Z the limit b ! 0.8 So we get d b Z d d 2 Z 2 ~x d 1 ds 1 ds(e sTzz es : Tzz :) z (2.26) d e i = 2 e s ( s) 2 (s) (2.25) The delta function term above can be dropped since this term does not contribute in 0 1 Stack = 2 b Z dd 2~x T1 1 : 8Actually, rather than drop this term, let us add it to a stack: We will update Stack everytime we nd a term of this type in our calculation. coordinates are de ned as x = x0 x1), we obtain the vector eld back to real time, i.e. z ! + and z ! Naively, it might seem that all the terms on the right hand side vanish in the b ! 0 limit. In fact, the terms involving Tzz do indeed vanish in this limit.9 However, the terms involving Tzz and Tzz get an enhancement from the s integral, coming from the s ln b and s ln b limits respectively. Taking the limit b ! 0 and Wick rotating (where the light-cone Z d ~x dx+ +T++(x+; x 0 + (2.27) where note that the rst term on the right hand side is integrated over the future Rindler horizon H+, while the second term is integrated over the past Rindler horizon H , shown in gure 1. (ii) Branch cut : now we come to the second remaining term supported over @Re+ [ @Re . Once again, deforming the s contours to get rid of the dependence from the stress tensors, we obtain = Z d ~x dx1 ds where we have also shown the Wick rotated Rindler coordinates. The hole region Rb corresponds to r < b and since we are again working in a region close to the 9By this we mean that the Tzz terms do not contribute to matrix elements in the class of states (1.15) which are of interest here. On the other hand, if we were to evaluate matrix elements in Rindler eigenstates we would nd potential divergences in this limit. Note that since we used a spectral representation for A0 at an intermediate stage, we were exactly evaluating this in Rindler eigenstates. So the order in which this limit is taken is a somewhat delicate issue which is best ignored on a rst pass. In appendix A we confront this issue explicitly. where t = @x0 , and the stress tensor is evaluated on the region A0, i.e. T E T (x0E = 0; x1; ~x) above. The rst term inside the brackets comes from @Re+ while (after the contour deformation s ! s + 2 It is clear from equation (2.28) that the s integral precisely picks out the double-pole at s = 0. A straightforward application of the residue theorem gives ~x dx1 t b T (0; x1; ~x); KA0 (iii) Inside the hole: we can follow the same methods as in (i). Pick coordinates close to the entangling surface such that: ds2 = dr2 + r2d 2 + d~x2 ! dr2 r2ds2 + d~x2 b 1 4 sinh2 s+i 1 2 + 1 4 sinh2 s i 2 ! t A0is : T : iAs0 (2.28) (2.29) (2.30) entangling surface we can take the di eomorphism at leading order to be independent of r. After shifting the integration contour s ! s + i and Wick rotating the eld we have: is+ 2 A0 Z d Z d x ds is 2 A0 1 The light-like coordinates where the stress tensor on the right hand side above is located are x = re s. We still have to integrate (2.31) over: : : : = Z d d 2 Z ~x I drr d r<b 1 ds 1 dependence is the same as in (2.25) and we can again do the integral. After ignoring the (s) contribution which vanishes in the limit b ! 0,10 this has exactly the e ect of switching the angular integral in the Euclidean calculation to a real time integral localized near the Rindler horizon: 0 < r < b and (see gure 4). The integrand is the stress tensor coupled to a real time di eomorphism 1 < s < 1 of the metric for the following vector eld: Kb = 2 Z 0<r<b ddx T ( x0) We have again ignored a contribution localized at x0 = 0, coming from the derivative of the step functions above, which vanishes in the limit b ! 0.11 It is not hard to see that (2.33) should vanish in the limit b ! 0. However it is somewhat enlightening to go another route and instead integrate by parts on (2.33). We get two terms, one from the r = b boundary and the other from precisely the past and future Rindler horizons on the boundary of the domain of dependence of A0. It turns out the former term cancels (2.26) prior to taking the b ! 0 limit (although we always need b small), and the later term is exactly the desired result given in (2.27) . So in the end when we add all the terms together, no b ! 0 limit is necessary and the null 10Once again, we add this term to the stack de ned in footnote 8: Stack ! Stack 2 Z dd 2~x Z b 0 11These terms go into the stack as well: Stack ! Stack + 2 Z dd 2~x Z b dx1(T10 0 + T00 1) = 2 0 Z dd 2~x Z b 0 + 2T10 0 + (T11 + T00) 1 where in the second equality we have integrated by parts; this is then exactly the extension of the x1 integral in (2.29) so that it ranges from 0 to 1. Even though all these terms vanish as b ! 0, it is satisfying that they add up like this. energy operators in (2.27) simply emerge. This is perhaps not too surprising since the r = b surface is imaginary, and there should be no dependence on b, however we nd the detailed cancelations that occur and the form in (2.33) intriguing (including in the running footnote Stack), perhaps hinting that there is a di erent way to do this calculation directly in real times. To summarize, putting everything together, we nd that the modular Hamiltonian over the deformed half-space is given by HJEP09(216)38 U KAU y = KA0 2 which is the result claimed in (1.14).12 We emphasize once again that the appearing in the second and third terms above are de ned at the entangling surface and in particular do not depend on the null coordinates x along the Rindler horizons. We can also derive a similar expression for the modular Hamiltonian corresponding to the complement Ac V KAc V y = KAc0 + 2 +T++ 2 + 2 A0c t T ; KAc0 (2.36) where H c are the Rindler horizons corresponding to the complement Ac0, and V : hAc ! hAc0 is a unitary transformation. Finally, putting these together, we obtain the following formula for the full modular Hamiltonian U KbAU y = KbA0 2 where we have de ned the light sheets L transformation given by U = U 12Roughly speaking, the \null-energy" terms measure the amount of modular energy leaving the Rindler wedge, while the commutator term comes from the action of the unitary transformations on the original (undeformed) modular Hamiltonian. Z H+ Z c H+ Z L+ Z H Z H c Z L = H A0 t Z Z T T + 2 t h T ; KbA0 i (2.37) [ Hc , and U : h ! h is a unitary In this section, we will consider the expectation value h jKbAj i in the class of states (1.15). We will then use the positivity of the operator KbA0 KbA to prove the averaged null energy condition within this class. For completeness, we begin with a brief review of the argument that KbA0 KbA is a positive operator, following [22].13 Consider any two states, which we take here to be the vacuum j0i and a non-trivial pure state j i. Given an entangling region A0 and the corresponding reduced density matrices A0 and A0 , one de nes the relative entropy (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) gion A0. tive quantity Relative entropy has a number of interesting properties. For instance, it is a posiS( A0 jj A0 ) 0 Further, if we pick another region A such that A A0 (more precisely, if D(A) D(A0), where D(A) is the domain of dependence of A) then the monotonicity of relative entropy implies S( Ajj A) S( A0 jj A0 ) Intuitively, the relative entropy measures the distinguishability between two states. From this point of view, the monotonicity property states that the distinguishability between two states decreases as we consider their reduced density matrices over smaller and smaller regions.14 From equations (3.2) and (3.4), we obtain S( A0 jj A0 ) = TrA0 A0 ln A0 TrA0 A0 ln A0 = hTrA0 A0 KA0 + hTrA0 TrA0 ( A0 KA0 ) i A0 ln A0 TrA0 ( A0 ln A0 ) i hKA0 i SEE[A0]: where KA0 is the modular Hamiltonian corresponding to the vacuum state over the rehKAi KAc0 hKA0 i hKAc i SEE[A] + SEE[Ac0] + SEE[A0] SEE[Ac] 0 0 where all modular Hamiltonians are de ned relative to the vacuum. Adding the two inequalities we have D 13Similar arguments have been used in [4, 50]. A rigorous proof of the positivity of this operator can also be found in [21] which uses methods of algebraic QFT. replica trick. 14See [51{53] for eld theoretic calculations of relative entropy in excited states, using a version of the Now, since all vacuum contributions vanish, we can drop the annihilates the vacuum for any region A). This implies . (This is because KbA D where the last equality follows from the purity of j i. Since this is true for any pure state j i, we deduce that KbA0 KbA is a positive operator. We now explicitly compute the expectation value of KbA0 KbA in the state where as before we take A0 to be the half-space x1 > 0, and A to be the deformed half-space. Using the relation we nd where h jKbAj i = TrA AKA TrAc Ac KAc h j(KbA0 KbA)j i = T( 1 ) + T( 2 ) where note that the unitary transformations U and V have dropped out inside the trace. The second term above is straightforward to evaluate from equations (2.35) and (2.36): Z L+ T( 2 ) = 2 +hT++i 2 where we have de ned h T i + 2 h jY j i h j i : Z t h h KbA0 ; T i i The rst two terms in (3.14) are the appropriate null energy expectation values (albeit integrated along the transverse ~x directions with arbitrary coe cients (~x)) which enter in the averaged null energy condition. We will see below that the last term in (3.14) is precisely cancelled by a contribution coming from T( 1 ). Consider for instance, the rst term in T( 1 ); from equations (2.9) and (3.12) we obtain Rd, and recall the notation On the right hand side, the correlators h i indicate Euclidean correlation functions on A0 Z L hY i X = X (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) Note that the Euclidean correlation function appears naturally from the path integral construction of the deformed density matrix for the excited state | we need only add an insertion of KA0 along A0 and trace | resulting in the above correlation function. Because of the KA0 operator insertion, we should remove an in nitesimal cut running along A0 from the region over which we integrate the di eomorphism R. Bearing this in mind, we can integrate by parts in the region R to obtain only a contribution from above and below the KA0 operator insertion, yielding a commutator: TrA0 vanishes at the locations of the operators O , which allows us to drop the divergence of the stress tensor. For the full modular Hamiltonian we then have: As promised, this term precisely cancels the last term in equation (3.14). We therefore conclude that Since (KbA0 hKbA0 i hKbAi = 2 +hT++i 2 h T + > 0 and < 0 by construction (see gure 1), the positivity of the operator KbA) leads to the averaged null energy conditions 1 Z L+ Z 1 Z 1 1 1 This concludes our proof that the monotonicity property of KbA implies the ANEC. While we presented the proof above in the context of half-spaces in Minkowski space-time, the above calculation can also be extended in an obvious way to general static bifurcate Killing horizons. In this case we would be studying the modular Hamiltonian for small deformations of the entangling cut away from the bifurcation point in the Hartle-Hawking state. The monotonicity constraint then leads to the ANEC for complete null generators of the Killing horizon. 4 Modular Hamiltonians in AdS/CFT In this section we make a connection between our results and the recently proposed JLMS formula [6]: KACFT = 4GN + O(GN ) (4.1) where A denotes the boundary subsystem, and M denotes the bulk region enclosed by the minimal/extremal Ryu-Takayanagi/HRT [54, 55] surface @MnA (which ends on the entangling surface @A, and is homologous to A). Further, the denote local terms on the extremal surface, which will not be relevant in the following discussion. The result (4.1) arises as a consequence of the formula for quantum corrections to the Ryu-Takaynagi entropy [56, 57] (see also [58].) Here, we want to perform a simple consistency check of our formula (2.37) for the full modular Hamiltonian of deformed half-spaces against equation (4.1). In particular, restricting to pure states so that the bulk area operator Area@MnA evaluates the same over A and Ac, (4.1) allows us to equate the full modular energies between the boundary and the bulk theories KbACFT = Kb Mbulk + O(GN ) (4.2) HJEP09(216)38 where note that the local terms on the extremal surface have also dropped out. In order to make contact with our previous results, we take A to be a small deformation of the boundary half-space A0 = fx1 > 0; x0 = 0g. If we use the coordinates z; x0; x1; ~x on the Poincare patch of AdS, with (4.3) (4.4) (4.5) gAdS = dz2 (dx0)2 + (dx1)2 + (d~x)2 z2 ; then the corresponding (undeformed) extremal surface in the bulk is given by the codimension two surface x 0 = x 1 = 0, and we have the corresponding undeformed region M0 = fx1 > 0; x0 = 0g. To linear order in the CFT shape deformation , we then expect h KbAC0FTi CFT = h bulkKb Mbu0lki bulk + O(GN ); where we have evaluated the deformations in the CFT and bulk modular Hamiltonians in an excited boundary state j iCFT and the dual bulk state j ibulk respectively. Further, bulk is the deformation in the bulk minimal surface as a consequence of the boundary shape deformation, and approaches in the limit z ! 0; it is xed away from the boundary by the requirement that the deformed bulk surface remain extremal [55]. For simplicity, we focus on null deformations on the CFT side, i.e. = 0. The left-hand side of equation (4.4) has been computed previously; see equation (3.20). Furthermore, if we view the bulk e ective theory as a weakly coupled quantum eld theory in background AdS geometry, then we expect that our at space arguments from the previous sections can be extended to the bulk e ective theory | this is because we have a Killing vector eld in AdS which generates a boost around the unperturbed extremal surface in the (x0; x1) plane. The bulk modular Hamiltonian then satis es a covariant version of (3.20): Z p h bulkKb Mbulk i bulk = h b+ulk(z; ~x)hT+bu+lk(x+; z; ~x)i bulk; where hij is the induced metric on the undeformed minimal surface @M0: h = dz2 + (d~x)2 z2 : A consequence of JLMS combined with our calculations is therefore: Z d d 2~xdx+ + h CFTjT+C+FTj CFTi = Z hdd 2~xdx+dz b+ulkh bulkjT+bu+lkj bulki (4.6) Our goal now is to establish this equivalence using the usual rules of AdS/CFT [59{61]. In particular in order to apply the JMLS argument the state under consideration must have a perturbative in GN back-reaction on the bulk AdS space-time and so we will consider boundary states of the form j CFTi = O(x )j0iCFT, where O is a single trace primary scalar operator of dimension = O( 1 ) in terms of large-N counting. The operator is = 0). In this case the dual bulk state can be identi ed as j bulki = limz!0 z (z; x )j0iAdS for the corresponding bulk eld . The equality in (4.6) can already be read o from the work of Hofman and Maldacena [39] but here, for completeness, we will give another derivation. For a CFT with a weakly coupled Einstein gravity dual in the bulk, the l.h.s. can be computed using Witten diagrams. In particular, we assume that the relevant part of the tensor in O(1=N ). We thus have for the integrand: hOy(x? )T+C+FT(x)O(x )i = T bulk ( ) Z Z m2 2 is the leading order bulk stress n = dzddy p GD++(z; x0; x)T bulk D (z; x0; x ) p 1 2 (4.8) (4.9) o (4.10) (4.11) bulk Lagrangian takes the form: S bulk Z AdSd+1 p G 1 GN The leading order Witten diagram contribution to the l.h.s. of (4.6) comes from the coupling between the graviton (h ) and the scalar stress tensor: (RG + ) + G where D (z; x0; x) and D (z; x0; x ) are the bulk-to-boundary propagators for the graviton and scalar respectively. We identify the products of scalar boundary-to-bulk propagators as giving rise to the expectation value of the bulk stress tensor in the state j ibulk: T bulk D (z; x0; x ) = hT bulk(z; x0)i bulk in T bulk( ). When viewed as a bulk operator, its expectation value in the bulk state j bulki is given by: h (z; x0)i bulk = lim ; 0!0 0 (z; x0) ( ; x ) (4.12) i The leading order (disconnected) diagram of the bulk 4-point function is given by products of bulk-to-bulk propagator:15 i i = @ Dbulk-to-bulk( ; x? ; z; x0)@ Dbulk-to-bulk(z; x0; 0; x ) The boundary-to-bulk and bulk-to-bulk propagators are related by the limit: !0 D (z; x0; x) = lim Dbulk-to-bulk(z; x0; ; x) (4.13) tions hold for the other two terms in T bulk and we conclude that: (z; x0)i bulk . Similar relaZ hOy(x? )T+C+FT(x+; xi)O(x )i = GD++(z; x0; x)hT bulk(z; x0)i bulk (4.14) Z Z We need to integrate this relation over R dx+dd 2~x +(xi) on the boundary. In particular, since +(~x) is independent of x+, we can take it out of the null integral, and replace R dx+D++(z; x0; x+; ~x) = ++Dshock(z; ~x0; ~x) (x0 ), where Dshock(z; ~x0; ~x) is the boundaryto-bulk propagator for the shock wave graviton mode: h++(z; x0 ; ~x) = f (z; ~x0) (x0 ). In AdSd+1, Dshock(z; ~x0; ~x) is determined by solving Einstein's equations for this metric uctuation giving the shock-wave equation:16 Dshock(z; yi; xi) = 0; lim Dshock( ; yi; xi) ! yi) The factor ++ (x0 ) localizes the bulk integral onto L+(@M0), and projects onto the (++) component of bulk stress tensor: +(xi)hT+C+FT(x+; xi)i CFT = ph ~+(z; ~x0)hT+bu+lk(z; x0 = 0; x0+; ~x0)i bulk ~+(z; ~x0) = z 2 d d 2 ~x +(~x)Dshock(z; ~x0; ~x) One can nally check from (4.15) that ~+(z; ~x0) satis es the extremal bulk extension of d 1 z ~+(z; ~x0) = 0; lim ~+( ; ~x0) ! +(~x0) !0 which is precisely what de nes b+ulk(z; ~x), making (4.16) equivalent to (4.6), consistent with JLMS formula. 15We analytically continue these propagators to real time such that the ordering is the appropriate one for computing expectation values in the state j ibulk. not used this fact. 16This shock wave metric is actually a full non-linear solution to Einstein's equations although we have (4.15) (4.16) (4.17) In this paper, following the circle of ideas in [16, 22, 50], we established a relation between the monotonicity of relative entropy and the averaged null energy condition in arbitrary QFTs, and in so doing proved the most general Hofman-Maldacena bounds on the data in CFT three-point functions. We will now summarize the perturbative calculation we performed to establish this connection and then conclude with possible future work. The general goal was to study perturbatively the shape dependence of modular Hamiltonians/energies. We did this by applying \perturbation theory for reduced density matrices" which turns out to have some novel features which we describe now. Schematically the important term in our calculation (2.15) came from expanding the log used to de ne the modular Hamiltonian. Here we give an alternative description of this expansion: ln A0 (1 + 1 A0 ) = KA0 1 X( 1 )n Bn h n=0 n! | h KA0 ; KA0 ; : : : KA0 ; A0 h n {tizmes 1 iii } +O( 2 ) (5.1) where Bn are the Bernoulli numbers. The right hand side comes about due to the non commutativity of the two matrices in the log on the left hand side. That is, these are the usual nested commutator terms in the Baker-Campbell-Hausdor formula keeping only terms to order O( ) (see also [62] for related discussion). This set of nested commutators clearly has something to do with the evolution with respect to KA0 - or in other words modular ow. So it should come as no surprise that these terms can be re-summed into an integral over is=2 A0 1 A0 is=2 A0 multiplied by some kernel - a fact we used in (2.17). In fact, in going from equation (5.1) to (2.17), one simply uses the following integral representation of the Bernoulli numbers [63, 64]:17 Bn = ( i)n Z 1 ( 2 )n 1 ds s n 4 sinh2( s+2i ) (n 2 Z): (5.2) Surprisingly this integral and kernel as well have the e ect of switching the original Euclidean di eomorphism contained in and used to move around the entangling surface, to a real time di eomorphism determined by the new vector eld given in (2.33). From here the null energy operators involved in the ANEC just pop out as boundary terms when integrating by parts over the real time di eomorphism. Of course in real times now a new boundary has opened up; what previously was the co-dimension 2 entangling surface at the origin in Euclidean space becomes a null hypersurface along the Rindler horizon where the null energy operators are de ned. The non-commutativity emphasized in (5.1) was of fundamental importance to our calculation. We feel that we do not fully understand the magic behind this calculation and that there are new surprises lurking if we go to higher orders in perturbation theory and try to systematize this approach. Similar tools were applied in [7, 12] to entanglement entropy 17Note that we pick the convention where B1 = + 12 ; also recall that B2m+1 = 0 for m = 1; 2 The corresponding terms in the integral representation (5.2) pick out the residue at s = 0, which is only non-trivial for n = 1. where it was important to control these commutator terms in order to nd agreement between this perturbative approach and known results from AdS/CFT. Here we have also established a similar agreement with AdS/CFT and in particular the recent proposal by JLMS [6] for the modular Hamiltonian in AdS/CFT. Apart from gaining a deeper understanding into the inner working of these calculations we now give some detail of future work that we think would be valuable to pursue. Sharpening the argument In the main sections of the paper our derivation eschewed any issues related to the precise de nition of entanglement and modular energy in quantum eld theory. Indeed these quantities are expected to be a icted by signi cant UV divergences, and possibly even ambiguities related to how one splits the degrees of freedom between A and Ac. Thus in order to calculate these quantities we must specify a regulator and a prescription for splitting the Hilbert space. However it became clear to us that we never needed to do this, and so any real discussion of a regulator was relegated to appendix A. Ultimately this should not have come as a surprise, the nal goal was to calculate either relative entropy or the full modular Hamiltonian - both of which are expected to be UV nite quantities and both of which can actually be given a de nition directly in the continuum [47, 48]. This de nition however was not convenient for our current calculation so at an intermediate step we needed to calculate the expectation of the full modular Hamiltonian in terms of the UV sensitive (half) modular Hamiltonian. Since we never explicitly saw these UV divergences, our manipulations should be regarded as formal.18 Appendix A is an attempt to remedy this, by giving some details of a brick wall like regulator [66] that renders the modular energy and associated quantities well-de ned. The brick wall regulator introduces dependence on the boundary conditions one chooses for elds at the wall close to the entangling surface. The regulated version of relative entropy does not satisfy the property of monotonicity (for a nite but small cuto scale) since the brick wall cuto is a rather drastic modi cation to the theory that does not allow one to compare di erent spatial regions with the same modi cation. So to claim a completely rigorous proof of the ANEC we still need to show that when we remove the brick wall cuto the quantity we get is the continuum version of relative entropy - which is then known to be monotonic [47]. This requires methods that are beyond the scope of this paper, and we leave this to future investigations. Ultimately we would like a mathematically rigorous derivation, perhaps without reference to density matrices and using methods of algebraic quantum eld theory [21, 48]. Finally we would like to understand if there are any restrictions on the state in which we calculate the expectation value of the deformed modular Hamiltonian. For example we formulated our state in terms of a local operator insertion at x su ciently general for a CFT. More generally, say for relativistic theories, our argument will go through relatively unmodi ed if we just insert a general state of the theory and 0 E = , which is 18They might be regarded as about as formal as the usual derivation of the replica trick for Renyi entropies in terms of a partition function on a singular surface [65]. its conjugate in at space along the Euclidean time slices x . However we are required to separate the di eomorphism that moves around the entangling surface away 0E = . We can make the region in which the di eomorphism acts small but we should be limited by j j the size of the di eomorphism vector eld at the entangling surface. This presumably puts some restriction on the state such that the expectation value of the stress tensor cannot get arbitrarily large. For example if we work with the state created by that the perturbative expansion converge and we can always arrange this to happen by 1 < j j 1. This is likely just the restriction a local operator insertion j RH h T i j taking a small enough spatial deformation. One obvious generalization involves attempting to prove the ANEC in other space-times as well as along more general complete achronal null geodesics.19 Along these lines it might be an easier rst step to try to apply the methods of this paper to stationary but not static black holes with the null generator lying along a bifurcate Killing horizon (like the Kerr black hole). Since we used the framework of perturbation theory starting from a state described by a known density matrix (the Hartle-Hawking state) we are not very optimistic this will succeed when we don't have such a starting point. Instead perhaps a more fruitful direction to pursue would be to consider the generalized second law (GSL) for quantum elds outside of a black hole with a static bifurcate Killing horizon. Here we are referring to the work of [16] where the GSL was proven for free as well as super renormalizable QFTs.20 The GSL applies to the following generalized entropy: Sgen = Area(@A) 4GN + SEE( A ) (5.3) (5.4) (5.5) HJEP09(216)38 19These are geodesics where no two points on the curve are timelike separated. The ANEC is known to fail in curved space-times where the null geodesic is chronal [37, 67]. 20The Hawking area theorem proves the GSL when the area term dominates in the GN ! 0 limit - that is for a classical dynamical background where the classical matter satis es the NEC. As discussed in [16], what remains, is to prove the GSL when classically the area does not increase - for quantum elds on a stationary black hole background plus free gravitons. For obvious reasons here we then focus on the static case, and leave out gravitons for simplicity. where Area(@A) refers to the area of a codimension-1 slice of the Killing horizon where the spatial region A ends (@A) and SEE is the entanglement entropy of the quantum elds outside this horizon slice. Applying the monotonicty of relative entropy to SEE( A Sgen Area(@A) 4GN + where now is a nite null deformation ( x+ = +(~y)) of the entangling surface @A to the future of the bifurcation surface @A0. The change in the area is simply due to the perturbative back reaction of the quantum elds on the space-time via Einstein's equations: d d 2 ~x Z 1 + dx+x+ hT++i + where we have made use of the Raychaudhuri equation with the correct future boundary condition appropriate for a causal horizon. To make further progress we need some handle on KA for general null deformations away from A0. This does not sound very promising for our perturbative approach, however it does seem like we can carry out our calculations to arbitrary orders in + [68]. Thus with some luck we might be able to prove a statement about KA and get a handle on (5.5) and possibly show the GSL in this case, Sgen 0. A further hint comes actually from AdS/CFT. For a Rindler space cut we have carried out a more detailed calculation21 than that outlined in section 4 where we previously showed the equality between the null energy operators in the bulk and boundary. More generally one can show for nite null deformations, but small perturbations to the state (in the 1=N sense): 2 Z 4GN + + 2 T+C+FT Z d 2 p Z 1 ~x h + bulk dx+ x + + bulk T+m+atter (5.6) and our notation is the same as that in section 4, where for example bulk is the bulk HRT extremal surface corresponding to the deformation + on the boundary and M is the spatial region between this extremal surface and A on the boundary. Here the area term is the change in the area of the extremal surface due to the backreaction on the metric g in the state (via Einstein's equations.) Note that the extremal surface condition in pure AdS for nite null deformations remains a linear di erential equation that matches with the in nitesimal version (4.17) and so b+ulk is the same extension as that used in section 4. Now comparing this statement with that of JLMS [6] we could consistently identify the + modular Hamiltonian for nite null deformed regions as: KACFT =? 2 Z d ~x dx+ x + + + T+C+FT (5.7) M up to an additive constant, with a similar equation holding for the bulk region modular Hamiltonian Kbulk. This is certainly not a proof. We have made two guesses (for the bulk and the boundary) and shown them to be self-consistent. And in particular this only works for a special class of states that don't have a large back reaction on the bulk. Note that this last issue also plagued our comparison between the bulk and boundary for small deformations. We simply note here that our perturbative approach, when considered at higher orders, can possibly prove such a statement.22 Of course if (5.7) is true then the GSL follows trivially since the right hand side of the inequality in (5.5) just vanishes. 21This calculation has some overlap with [69] and the details will be reported elsewhere. 22Actually a simpler argument is to take the perturbative result we have derived for null shape deformations and then use the QFT boost generator around the original undeformed Rindler cut to amplify the deformation. This boost will then act on the state. However if we work in a su ciently general state this should not matter. This process seems to work, and agrees with (5.7), when trying to construct the full modular Hamiltonian and we leave the details of how this works for the half space modular Hamiltonian for the future. We thank Aron Wall for suggesting this argument to us. Finally we point out that in some sense these calculations have already been pushed to higher orders. Rather than consider the excited state modular energy, if we just calculate the modular Hamiltonian in the original vacuum state it should reproduce the entropy of the vacuum. At rst order this vanishes but the second order variation of entropy in a CFT was calculated in [7] using similar methods to this paper. This quantity is sometimes referred to as entanglement density [70]. Although it was not realized at the time the answer in [7] can be related to a correlation function of two \null energy operators" - the same null energy operators that appear in the (half sided) modular Hamiltonian in this work. This will be the subject of a forthcoming paper [71]. Taken together this hints at a unifying picture for vacuum entanglement in CFTs related to null energy operators that may even pave the way to a new understanding and proof of the Ryu-Takayanagi [54] and HRT [55] proposals for calculating entanglement entropy in the vacuum state of holographic CFTs. Acknowledgments It is a pleasure to thank Xi Dong, Gary Horowitz, Veronika Hubeny, Aitor Lewkowycz, Don Marolf, Mukund Rangamani, David Simmons-Du n and Aron Wall for discussions and suggestions. Work supported in part by the U.S. Department of Energy contract DE-FG02-13ER42001 and DARPA YFA contract D15AP00108. A Cuto at the entangling surface In this appendix we would like to give a prescription for regulating the modular energy that we calculate in the main part of the paper. We go through this in some depth because the arguments we gave previously were somewhat formal. Although the quantity in which we are ultimately interested | the full modular Hamiltonian | is UV nite [48], at intermediate steps we encountered quantities which are not. In particular the modular energy of some state is expected to have the same UV divergences as the entanglement entropy of that state because the di erence between them is the relative entropy which is UV nite.23 Thus the issues here are the same as the usual issues of de ning entanglement entropy in the continuum.24 There are several ways to de ne a regulated version of entanglement entropy, but the most convenient for us will be a \brick wall" regulator [66]. This is so we can still use Euclidean path integral methods to construct the density matrices in question. Apart from possible IR issues the entropies are now nite - the IR issues do not concern us and cancel when evaluating the di erences between excited and vacuum states, at least for states that are su ciently close to the vacuum near the boundaries of space. 23There are still several reasons to expect some of our intermediate steps to be nite. Any divergences should be local to the entangling surface, and assuming our regulator is geometric[72{74] no such term which respects the S(A) = S(Ac) purity condition can generate a divergence for rst order spatial shape deformations. Similarly there is a general expectation that any such divergences cancel in the di erence S( A) although we will nd evidence that this cancellation might not always occur. Of course these variations and di erences are still calculated in terms of divergent quantities so we proceed. 24For a recent discussion of some of the issues involved see [72, 75]. When the QFT in question is a gauge theory there are even questions about how the degrees of freedom are split between two spatial regions [76]. r = a 0 xE x 1 state. We cut out a cylindrical region of radius r = a around the entangling surface, with brickwall-like boundary conditions. Also shown is the ctitious cuto surface of radius r = b. Roughly speaking we can simply go through our calculation in sections 2 and 3.2 with the reduced density matrices de ned via path integrals on manifolds with a cylindrical region of radius a cut out from around the entangling surface: A0;g ! A0;g(a) (see gure 2). In order to to do this consistently we should impose boundary conditions on the cuto surface - we will assume that the boundary conditions at the cuto surface decouple in the limit a ! 0. We might also need to add new degrees of freedom here [77, 78] and there are good reasons to believe these should decouple when calculating such things as relative entropy [76]. We also require the following: Rotation/Boost invariance for the undeformed Rindler region. This is so that KA0 still has the interpretation as the generator of rotations/boosts around the cuto surface. For example this will require that the stress tensor at the cuto surface is constrained to have zero rotation should be required as part of the boundary conditions.25 ux T rjr!a ! 0 into the cuto cylinder. This For a more general region | we cut out a cylinder in Gaussian normal coordinates. Here of course we do not have rotation invariance. We use normal coordinates so we can still use the relation (2.5) derived in the main text. One way to do this is to pick the di eomorphism to map the deformed entangling surface to Gaussian normal coordinates | where the regulator is then picked to be a metric distance a orthogonal to the surface away from A. For us this amounts to the choices: 2 2 g dx dx = dzdz + (A.1) (A.2) 25Note however that this can fail in the case of chiral theories, in which case the boost symmetry is anomalous [79{82]. where we have expanded the di eomorphism and the metric close to the entangling surface. We then cuto the path integral which de nes A0;g(a) in (2.5) at r = jzj = a supplying some appropriate yet unspeci ed boundary conditions. After making this slight modi cations the di eomorphism acts the same way as in the bulk of the text in particular there is no boundary term due to the displacement of the cuto surface (although of course the new stress tensor could have delta function contributions on the cuto surface.) At the very minimum we require that for some local operator inserted in the path integral that de ned A0;g(a) we should have: a!0 lim TrA0 A0;g(a)O(x) = hO(x)i (A.3) (A.4) (A.5) Following the steps below (2.18) in section 2.2 for the change in modular Hamiltonian due to the stress tensor deformation, the di erences are due to a slightly modi ed di eomorphism and a di erent region of integration for the stress tensor R0 in the Euclidean plane which is cuto by r > a; see gure 2. This cuto is distinct from the imaginary cuto surface de ned in section 2.2 with r > b and we will take a b. Indeed splitting the contribution from the stress tensor integral into the three regions as we did previously, there is only one term which is sensitive to the cuto in the limit a b and this is the contribution from inside the hole a < r < b which we call Kb(a). The other two contributions from the branch cut and the imaginary cuto surface give identical results as in the main text. With the same set of manipulations we can write the potentially problematic term as: Kb(a) = 2 Z a<r<b ddx : T : ge where the resulting real time metric deformation is: e g dx dx = 1 2 dxidxj Note the integration region is now a section of a solid hyperboloid. This is slightly modi ed from (2.33) and (2.34) since we are now working in Gaussian normal coordinates. Of course to analyze the limiting properties of (A.4) we should trace it against our state Tr A0 ( ). At this point it is possible to remove the brick wall regulator a ! 0 from (A.4). Dependence on a appears in the integration region R0 as well as implicitly in T since this is the appropriate eld theory stress tensor in the presence of a boundary. Note that the boundary conditions on the brick wall in Euclidean space have naturally been mapped to Rindler space in real times along the hyperbola r = a, 1 < s < 1. The claim is that we can remove the regulator from the integrand using the requirement (A.3). Of course the remaining ddx integral may still be divergent, however we found this not to be the case in the main text. It is this sense in which we expect the boundary conditions on the surface r = a to decouple. To show this rigorously we would have to show the integrand converges su ciently uniformly to the a = 0 limit. Note that the metric deformation gij in Rindler coordinates and so this is a mild condition on the behavior of the stress tensor re jsj close to the brick wall.26 To say anything further we would need to specify more about the boundary conditions than we are willing to. However since the details of the boundary conditions are not important for de ning relative entropy or the full modular Hamiltonian in the continuum of a QFT, it must be the case that any divergences we might see here should cancel when calculating these nal quantities. Instead we can turn this condition around and demand that this should be true for any brick wall regulator that is supposed to be a good regulator for calculating modular energy. We now turn to the second contribution to the deformed modular energy (the rst term in (3.12).) Compared to our previously obtained expressions we now nd a contribution from the boundary of the cuto region at r = a which looks like: = a I TrA0 A0 KA0 n h T (x)KA0 i h T (x)i hKA0 i (A.6) If we instead calculate this contribution to the shape deformation of the full modular hamiltonian (this was de ned as T( 1 ) in (3.12)) we get a term coming from the complement Ac0 which adds to give the total contribution to T( 1 ) coming from the cuto surface @Ra: T( 1 ) I (x)i hKbA0 i (A.7) where we remind the reader that KbA0 is the undeformed full modular Hamiltonian. In the limit a ! 0, the above term appears to be linearly suppressed; however one might worry that there are potential enhancements from the stress tensor coming close to KbA0 in the rst term above. To see that this does not happen, recall that KbA0 is a conserved charge, namely the generator of rotations around the entangling surface. Consequently, we can freely move it away from cuto surface as well as the other stress tensor inside the above correlator. Here we have to take into account the fact that the boundary condition on we could move KbA0 o to x0E = the cuto surface should not allow for any KbA0 ux into the cuto surface Tr ! 0. If 1, then the corresponding term would vanish, because KbA0 annihilates the vacuum. However, as we keep moving KbA0 away from the stress tensor, we will eventually cross the operator insertions O or O y (depending on whether we move KbA0 towards x0 E ! contribution of the form h T 1 or x0 [KbA0 ; O m ] E ! +1). Every such crossing gives a non-trivial i, where denotes the remaining operator insertions. However, it should now be clear that these remaining terms are correlation functions between well-separated operators (as long as a), and we do not get any enhancement to cancel the factor of a. Therefore, we conclude that the contribution from the cuto surface to T( 1 ) vanishes in the limit a ! 0. We claim victory. Before moving on we note that if we did not do the subtraction that de ned the full modular energy, the term (A.6) might still be divergent. We can give the following crude estimate for any such divergence. Note that the half sided modular Hamiltonian, as an 26Note because of the :: vacuum subtraction for the stress tensor any divergence that might appear exactly at the cuto surface when a is xed (say due to an image charge) is state independent and will cancel. The potential divergence we are worried about is in the subsequent limit a ! 0. integral over the stress tensor, can still be moved around but now it is always anchored to the cuto surface. We can use this freedom to move the two stress tensors in (A.7) as far apart as possible - on opposite sides of the hole. To get an estimate we now replace the correlation function in the rst term of (A.6) with the CFT correlation function without surface - on at Euclidean space. We need to consider the OPE of two stress tensors schematically of the from: T T X C ( x2) d+ (A.8) where x 2 (a + r)2 + (~x ~x0)2 and where r > a refers to the location of the modular Hamiltonian stress tensor. Here O we can expand (~x) and hO (~x0)i possibly contribute a divergence. Close to the entangling surface for any divergent term are some local primary operators and only scalars can by taking ~x0 ~x. We get some leading term from the unit operator. But there are good reasons this term should vanish. Firstly it is state independent and so should occur for the vacuum state, but there is simply no term we can write down at linear order in which is local to the entangling surface and has the required rotation/boost invariance around the entangling surface. However now consider a non unit operator, we no longer have rotation around the entangling surface. Then using scale invariance we only expect a divergent contribution to the modular energy of the form: C a Z d d 2 Z a drr( x2) d+ =2+1=2 @ O (~x0) 2 d+ C d d 2 (A.9) Naively one would have expected that such a contribution is not possible for uniform deformations independent of ~x - since in that case we can write an expression for KA and KA0 in the absence of the cuto surface and there is seemingly no divergence. However since these are half sided modular Hamiltonians it seems we should have allowed for the possibility that even the un-deformed KA0 has a local divergence: KA0 2 d+ C + nite Z (A.10) At least this seems to be required if we want the answer to be consistent with our UV regulator and di eomorphism invariance. This calculation is far too crude to be trusted, but it does suggest that any kind of brick wall cuto leaves one susceptible to state dependent divergences in the half sided modular energy of the above nature. In the main text we could have taken such divergences into account simply by adding (A.10) and this would have generated the term in (A.9) without any need for a brick wall. This new divergent contribution occurs if there is a very relevant d 2 scalar operator appearing in the T T OPE. For example it cannot be charged under any symmetries. Symmetries would also disallow (A.10). 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Thomas Faulkner, Robert G. Leigh, Onkar Parrikar. Modular Hamiltonians for deformed half-spaces and the averaged null energy condition, Journal of High Energy Physics, 2016, 38, DOI: 10.1007/JHEP09(2016)038