Localized thermal states and negative energy

A bstractWe construct localized states defined in a ball or the half-space of a conformal field theory (CFT) in Minkowski that are thermal with respect to the local modular flow. We compute their energy density at arbitrary temperature for a variety of CFTs, and find values for which it is negative and divergent at the boundary. Despite this singular behavior we show that the energy measured by an observer is consistent with the bounds present in the literature. For holographic CFTs these states are captured by hyperbolic black holes in anti-de Sitter, where the negative energy in field theory amounts to the well known negative mass of the black hole. As a byproduct, we show that the Casini-Huerta-Myers proof of the Ryu-Takayangi holographic entanglement formula for the vacuum reduced to a ball can be naturally extended to include half-space regions.

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title: Quantum inequalities for the electromagnetic field
abstract: A quantum inequality for the quantized electromagnetic field is developed for observers in static curved spacetimes. The quantum inequality derived is a generalized expression given by a mode function expansion of the four-vector potential, and the sampling function used to weight the energy integrals is left arbitrary up to the constraints that it be a positive, continuous function of unit area and that it decays at infinity. Examples of the quantum inequality are developed for Minkowski spacetime, Rindler spacetime and the Einstein closed universe.

title: Modular energy inequalities from relative entropy
abstract: A bstractWe obtain new constraints for the modular energy of general states by using the monotonicity property of relative entropy. In some cases, modular energy can be related to the energy density of states and these constraints lead to interesting relations between energy and entropy. In particular, we derive new quantum energy inequalities that improve some previous bounds for the energy density of states in a conformal field theory. Additionally, the inequalities derived in this manner also lead us to conclude that the entropy of the state further restricts the possible amount of negative energy allowed by the theory.

title: Lectures on quantum energy inequalities
abstract: Quantum field theory violates all the classical energy conditions of general relativity. Nonetheless, it turns out that quantum field theories satisfy remnants of the classical energy conditions, known as Quantum Energy Inequalities (QEIs), that have been developed by various authors since the original pioneering work of Ford in 1978. These notes provide an introduction to QEIs and also to some of the techniques of quantum field theory in curved spacetime (particularly, the use of microlocal analysis together with the algebraic formulation of QFT) that enable rigorous and general QEIs to be derived. Specific examples are computed for the free scalar field and their consequences are discussed. QEIs are also derived for the class of unitary, positive energy conformal field theories in two spacetime dimensions. In that setting it is also possible to determine the probability distribution for individual measurements of certain smearings of the stress-energy tensor in the vacuum state.

title: Quantum coherence effects and the second law of thermodynamics
abstract: Negative energy densities and fluxes due to quantum coherence effects in quantum field theories are discussed. Such negative energy fluxes seemingly lead to a breakdown of the second law of thermodynamics. It is argued that such a breakdown will not in fact occur if a negative energy flux F is constrained by an inequality of the form │F│≲ T-2 where T is a characteristic time over which the negative energy flux occurs. Particular models in which negative energy fluxes occur are examined, and it is found that such an inequality is indeed satisfied. Quantum violations of the dominant energy condition, which requires that the local energy density be greater than or equal to the magnitude of the pressure are also discussed briefly.

title: Constraints on negative-energy fluxes.
abstract: Locally negative energy due to quantum coherence effects in quantum field theory is discussed. In a previous work, it was argued that a beam carrying negative energy must satisfy an uncertainty-principle-type inequality of the form \ensuremath{\Vert}\ensuremath{\Delta}E\ensuremath{\Vert}\ensuremath{\Delta}t\ensuremath{\le}1, where \ensuremath{\Vert}\ensuremath{\Delta}E\ensuremath{\Vert} is the magnitude of the negative energy that may be transmitted in a time \ensuremath{\Delta}t. This conclusion applied only to two-dimensional spacetime, and was based on an examination of particular classes of quantum states. In the present work, we give more precise formulations of this type of inequality for a free massless scalar field in both two- and four-dimensional flat spacetime. These inequalities are proven to hold for all quantum states. The physical interpretation of these inequalities is also discussed, and it is argued that they are likely to prevent negative energy from producing such large-scale effects as violations of the second law of thermodynamics or of cosmic censorship.

title: Wormholes, time machines, and the weak energy condition.
abstract: It is argued that, if the laws of physics permit an advanced civilization to create and maintain a wormhole in space for interstellar travel, then that wormhole can be converted into a time machine with which causality might be violatable. Whether wormholes can be created and maintained entails deep, ill-understood issues about cosmic censorship, quantum gravity, and quantum field theory, including the question of whether field theory enforces an averaged version of the weak energy condition.

title: Must time-machine construction violate the weak energy condition?
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title: Warp drive with zero expansion
abstract: It is commonly believed that Alcubierre's warp drive works by contracting space in front of the warp bubble and expanding the space behind it. We show that this contraction/expansion is but a marginal consequence of the choice made by Alcubierre and explicitly construct a similar spacetime where no contraction/expansion occurs. Global and optical properties of warp-drive spacetimes are also discussed.

title: Modular Hamiltonians for deformed half-spaces and the averaged null energy condition
abstract: A bstractWe study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on ℝ1,d−1$$ {\mathrm{\mathbb{R}}}^{1,d-1} $$. 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.

title: Averaged null energy condition from causality
abstract: A bstractUnitary, Lorentz-invariant quantum field theories in flat spacetime obey mi-crocausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, ∫ duTuu, must be non-negative. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to n-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form ∫ duXuuu···u ≥ 0. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.

title: A general proof of the quantum null energy condition
abstract: We prove a conjectured lower bound on 〈T__(x)〉
 ψ
 in any state ψ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through x. Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a “probe” operator 
 
 $$ \mathcal{O} $$
 
 O
 
 
 in the state ψ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality on modular time for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. We argue that the proof extends to more general relativistic QFT with an interacting UV fixed point and also prove a higher spin version of the QNEC. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point.

title: Light-ray operators in conformal field theory
abstract: A bstractWe argue that every CFT contains light-ray operators labeled by a continuous spin J. When J is a positive integer, light-ray operators become integrals of local operators over a null line. However for non-integer J , light-ray operators are genuinely nonlocal and give the analytic continuation of CFT data in spin described by Caron-Huot. A key role in our construction is played by a novel set of intrinsically Lorentzian integral transforms that generalize the shadow transform. Matrix elements of light-ray operators can be computed via the integral of a double-commutator against a conformal block. This gives a simple derivation of Caron-Huot’s Lorentzian OPE inversion formula and lets us generalize it to arbitrary four-point functions. Furthermore, we show that light-ray operators enter the Regge limit of CFT correlators, and generalize conformal Regge theory to arbitrary four-point functions. The average null energy operator is an important example of a light-ray operator. Using our construction, we find a new proof of the average null energy condition (ANEC), and furthermore generalize the ANEC to continuous spin.

title: Recovering the QNEC from the ANEC
abstract: We study the relative entropy in QFT comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy (ANE) of possible purifications. This variational expression can be used to easily prove the quantum null energy condition (QNEC). We formulate Wall’s conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this context. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.

title: Conformal collider physics: energy and charge correlations
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title: Holographic GB gravity in arbitrary dimensions
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title: Local modular Hamiltonians from the quantum null energy condition
abstract: The vacuum modular Hamiltonian $K$ of the Rindler wedge in any relativistic quantum field theory is given by the boost generator. Here we investigate the modular Hamiltoninan for more general half-spaces which are bounded by an arbitrary smooth cut of a null plane. We derive a formula for the second derivative of the modular Hamiltonian with respect to the coordinates of the cut which schematically reads $K" = T_{vv}$. This formula can be integrated twice to obtain a simple expression for the modular Hamiltonian. The result naturally generalizes the standard expression for the Rindler modular Hamiltonian to this larger class of regions. Our primary assumptions are the quantum null energy condition --- an inequality between the second derivative of the von Neumann entropy of a region and the stress tensor --- and its saturation in the vacuum for these regions. We discuss the validity of these assumptions in free theories and holographic theories to all orders in $1/N$.

title: Energy density from second shape variations of the von Neumann entropy
abstract: We compute the local second variation of the von Neumann entropy of a region in theories with a gravity dual. For null variations our formula says that the diagonal part of the Quantum Null Energy Condition is saturated in every state, thus providing an equivalence between energy and entropy. We prove that the formula holds at leading order in 1/N, and further argue that it will not be affected at higher orders. We conjecture that the QNEC is saturated in all interacting theories. We also discuss the special case of free theories, and the implications of our formula for the Averaged Null Energy Condition, Quantum Focusing Conjecture, and gravitational equations of motion. We show that the leading-order gravitational equations of motion, Einstein's equations, are equivalent to leading-order saturation of the QFC for Planck-width deformations.

title: Quantum Inequality Restrictions on Negative Energy Densities in Curved Spacetimes
abstract: In quantum field theory, there exist states in which the expectation value of the energy density for a quantized field is negative. These negative energy densities lead to several problems such as the failure of the classical energy conditions, the production of closed timelike curves and faster than light travel, violations of the second law of thermodynamics, and the possible production of naked singularities. Although quantum field theory introduces negative energies, it also provides constraints in the form of quantum inequalities (QI’s). These uncertainty principle-type relations limit the magnitude and duration of any negative energy. We derive a general form of the QI on the energy density for both the quantized scalar and electromagnetic fields in static curved spacetimes. In the case of the scalar field, the QI can be written as the Euclidean wave operator acting on the Euclidean Green’s function. Additionally, a small distance expansion on the Green’s function is used to derive the QI in the short sampling time limit. It is found that the QI in this limit reduces to the flat space form with subdominant correction terms which depend on the spacetime geometry. Several example spacetimes are studied in which exact forms of the QI’s can be found. These include the three- and four-dimensional static Robertson-Walker spacetimes, flat space with perfectly reflecting mirrors, Rindler and static de Sitter space, and the spacetime outside a black hole. In all of the above cases, we find that the quantum inequalities give a lower limit on how much negative energy may be observed relative to the vacuum energy density of the spacetime. For the particular case of the black hole, it is found that the quantum inequality on the energy density is measured relative to the Boulware vacuum. Finally, the application of the quantum inequalities to the Alcubierre warp drive spacetime leads to strict constraints on the thickness of the negative energy region needed to maintain the warp drive. Under these constraints, we discover that the total negative energy required exceeds the total mass of the visible universe by a hundred billion times, making warp drive an impractical form of transportation.

title: Quantum states with negative energy density in the Dirac field and quantum inequalities
abstract: Energy densities of the quantum states that are superposition of two multi-electron–positron states are examined. It is shown that the energy densities can be negative when two multi-particle states have the same number of electrons and positrons or when one state has one more electron–positron pair than the other. In the cases in which negative energy could arise, we find that the energy is that of a positive constant plus a propagating part which oscillates between positive and negative, and the energy can dip to negative at some places for a certain period of time if the quantum states are properly manipulated. It is demonstrated that the negative energy densities satisfy the quantum inequality. Our results also reveal that for a given particle content, the detection of negative energy is an operation that depends on the frame where any measurement is to be performed. This suggests that the sign of energy density for a quantum state may be a coordinate-dependent quantity in quantum theory.

title: The vacuum energy density for spherical and cylindrical universes.
abstract: The vacuum energy density (Casimir energy) corresponding to a massless scalar quantum field living in different universes (mainly no‐boundary ones), in several dimensions, is calculated. Hawking’s zeta function regularization procedure supplemented with binomial expansion is shown to be a rigorous and well suited method for performing the analysis. It is compared with other more involved techniques. The principal‐part prescription is used to deal with the poles that eventually appear. Results of the analysis are the absence of poles at four dimensions (4D) (for a 4D Riemann sphere and for a 4D cylinder of 3D Riemann spherical section), the total coincidence of the results corresponding to a 3D and a 4D cylinder (the first after pole subtraction), and the fact that the vacuum energy density for cylinders is (in absolute value) over an order of magnitude smaller than for spheres of the same dimension.

title: Quantum Inequalities on the Energy Densityin Static
abstract: Quantum inequality restrictions on the stress-energy tensor for negative energy are developed for three and four-dimensional static spacetimes. We derive a general inequality in terms of a sum of mode functions which constrains the magnitude and duration of negative energy seen by an observer at rest in a static spacetime. This inequality is evaluated explicitly for a minimally coupled scalar eld in three and four-dimensional static Robertson-Walker universes. In the limit of vanishing curvature, the at spacetime inequalities are recovered. More generally, these inequalities contain the eeects of spacetime curvature. In the limit of short sampling times, they take the at space form plus subdominant curvature-dependent corrections.

title: Stress tensors from trace anomalies in conformal field theories
abstract: Using trace anomalies, we determine the vacuum stress tensors of arbitrary even dimensional conformal field theories in Weyl flat backgrounds. We demonstrate a simple relation between the Casimir energy on the real line times a sphere and the type A anomaly coefficient. This relation generalizes earlier results in two and four dimensions. These field theory results for the Casimir are shown to be consistent with holographic predictions in two, four, and six dimensions.

title: Universal entanglement and boundary geometry in conformal field theory
abstract: A bstractEmploying a conformal map to hyperbolic space cross a circle, we compute the universal contribution to the vacuum entanglement entropy (EE) across a sphere in even-dimensional conformal field theory. Previous attempts to derive the EE in this way were hindered by a lack of knowledge of the appropriate boundary terms in the trace anomaly. In this paper we show that the universal part of the EE can be treated as a purely boundary effect. As a byproduct of our computation, we derive an explicit form for the A-type anomaly contribution to the Wess-Zumino term for the trace anomaly, now including boundary terms. In d = 4 and 6, these boundary terms generalize earlier bulk actions derived in the literature.

title: On the vacuum stress induced by uniform acceleration or supporting the ether
abstract: The stresses induced in the vacuum by the uniform acceleration of an infinite plane conductor are computed for the massless scalar and electromagnetic fields. Both Dirichlet and Neumann boundary conditions are considered for the scalar field; far from the conductor it is found, independently of the boundary condition, that the vacuum stress is ‘local’ and corresponds to the absence from the vacuum of black body radiation. Approaching the conductor, the energy density in the Dirichlet case is slightly lower than the ‘local’ term, and in the Neumann case slightly higher. At very small distances it again has the same asymptotic form for both scalar fields. For the electromagnetic field the results are similar to those for the scalar field with Dirichlet boundary conditions. Far from the conductor the spectrum is again black-body, though not Planckian. In all cases the acausal nature of ‘ perfect conductor ’ boundary conditions prevents the stress tensor from being finite on the conductor.

title: Fermion fields in accelerated states
abstract: The massless spin-½ and spin-3/2 fields are quantized in the ‘Rindler wedge.’ The vacuum expectation value of the stress-energy tensor is calculated for the spin-½ field and is found to correspond to the absence from the vacuum of black body radiation. Though thermal, the spectrum of the stress tensor has a non-Planckian form.

title: Radiation from a moving mirror in two dimensional space-time: conformal anomaly
abstract: The energy-momentum tensor is calculated in the two dimensional quantum theory of a massless scalar field influenced by the motion of a perfectly reflecting boundary (mirror). This simple model system evidently can provide insight into more sophisticated processes, such as particle production in cosmological models and exploding black holes. In spite of the conformally static nature of the problem, the vacuum expectation value of the tensor for an arbitrary mirror trajectory exhibits a non-vanishing radiation flux (which may be readily computed). The expectation value of the instantaneous energy flux is negative when the proper acceleration of the mirror is increasing, but the total energy radiated during a bounded mirror motion is positive. A uniformly accelerating mirror does not radiate; however, our quantization does not coincide with the treatment of that system as a ‘static universe’. The calculation of the expectation value requires a regularization procedure of covariant separation of points (in products of field operators) along time-like geodesics; more naïve methods do not yield the same answers. A striking example involving two mirrors clarifies the significance of the conformal anomaly.

title: Quantum radiation by moving mirrors
abstract: The quantum radiation of scalar particles by a moving mirror is discussed. A perturbation method is developed which allows one to calculate the radiation produced by a boundary of arbitrary shape in the nonrelativistic limit. It is applied to the case of a mirror in two-dimensional spacetime and reproduces the results of Fulling and Davies. It is also applied to the case of a plane mirror in four-dimensional spacetime, and a formula is obtained for the total energy radiated by an arbitrary nonrelativistic motion.

title: Moving mirrors, black holes, and cosmic censorship.
abstract: We examine negative-energy fluxes produced by mirrors moving in two-dimensional charged-black-hole backgrounds. If there exist no constraints on such fluxes, then one might be able to manipulate them to achieve a violation of cosmic censorship by shooting a negative-energy flux into an extreme $Q=M$ or near-extreme Reissner-Nordstr\"om black hole. However, if the magnitude of the change in the mass of the hole $\ensuremath{\Delta}M$, resulting from the absorption of this flux, is small compared to the normal quantum uncertainty in the mass expected from the uncertainty principle $\ensuremath{\Delta}E\ensuremath{\Delta}T\ensuremath{\ge}1$, then such changes should not be macroscopically observable. We argue that, given certain (physically reasonable) restrictions on the trajectory of the mirror, this indeed seems to be the case. More specifically, we show that $|\ensuremath{\Delta}M|$ and $\ensuremath{\Delta}T$, the "effective lifetime" of any naked singularity thus produced, are limited by an inequality of the form $|\ensuremath{\Delta}M|\ensuremath{\Delta}Tl1$. We then conclude that the negative-energy fluxes produced by two-dimensional moving mirrors do not lead to a classically observable violation of cosmic censorship.

title: Local quantum physics : fields, particles, algebras
abstract: This textbook gives a comprehensive account of local quantum physics understood as the synthesis of quantum theory with the principle of locality. Centered on the algebraic approach, it describes both the physical concepts and the mathematical structures and their consequences. These include the emergence of the particle picture, general collision theory covering the cases of massless particles and infraparticles, the analysis of possible charge structures and exchange symmetries including braid group statistics. Thermal states of an unbounded medium and local equilibrium are discussed in detail. The author describes both the ideas and to give a critical assessment of future perspectives.

title: APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory
abstract: The 2016 APS Medal for Excellence in Physics was given to Edward Witten. This contribution was invited in conjunction with this award. These original notes contain concise explanations of some key results in the axiomatic and algebraic approaches to quantum field theory, which are relevant to quantum entanglement. They serve to put the connection between quantum field theory and quantum information theory on a precise and rigorous footing.

title: Entanglement Measures and Their Properties in Quantum Field Theory
abstract: An entanglement measure for a bipartite quantum system is a state functional that vanishes on separable states and that does not increase under separable (local) operations. It is well-known that for pure states, essentially all entanglement measures are equal to the v. Neumann entropy of the reduced state, but for mixed states, this uniqueness is lost. In quantum field theory, bipartite systems are associated with causally disjoint regions. There are no separable (normal) states to begin with when the regions touch each other, so one must leave a finite "safety-corridor". Due to this corridor, the normal states of bipartite systems are necessarily mixed, and the v. Neumann entropy is not a good entanglement measure in the above sense. 
In this paper, we study various entanglement measures which vanish on separable states, do not increase under separable (local) operations, and have other desirable properties. In particular, we study the relative entanglement entropy, defined as the minimum relative entropy between the given state and an arbitrary separable state. We establish rigorous upper and lower bounds in various quantum field theoretic (QFT) models, as well as also model-independent ones. The former include free fields on static spacetime manifolds in general dimensions, or integrable models with factorizing $S$-matrix in 1+1 dimensions. The latter include bounds on ground states in general conformal QFTs, charged states (including charges with braid-group statistics) or thermal states in theories satisfying a "nuclearity condition". Typically, the bounds show a divergent behavior when the systems get close to each other--sometimes of the form of a generalized "area law"--and decay when the systems are far apart. Our main technical tools are of operator algebraic nature.

title: Holography of negative energy states
abstract: Quantum states with negative energy densities have been long known to exist in quantum field theories. We explore the structure of such states for holographic theories using quantum information theory tools and show how certain negative energy states are naturally captured by the thermodynamics of black holes with hyperbolic horizon at zero temperature, suggesting that they provide a dual description of those states. Our results give a satisfying field theory understanding of the distinct thermodynamics of such black holes.

title: Towards a derivation of holographic entanglement entropy
abstract: We provide a derivation of holographic entanglement entropy for spherical entangling surfaces. Our construction relies on conformally mapping the boundary CFT to a hyperbolic geometry and observing that the vacuum state is mapped to a thermal state in the latter geometry. Hence the conformal transformation maps the entanglement entropy to the thermodynamic entropy of this thermal state. The AdS/CFT dictionary allows us to calculate this thermodynamic entropy as the horizon entropy of a certain topological black hole. In even dimensions, we also demonstrate that the universal contribution to the entanglement entropy is given by A-type trace anomaly for any CFT, without reference to holography.

title: Holographic Derivation of Entanglement Entropy from AdS/CFT
abstract: A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdSd+2, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS3. We also compare the entropy computed in AdS5×S 5 with that of the free N = 4 super Yang-Mills.

title: Aspects of Holographic Entanglement Entropy
abstract: , where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.

title: Entanglement entropy for even spheres
abstract: The coefficient of the logarithmic term in the entropy on even spheres is re-computed by the local technique of integrating the finite temperature energy density up to the horizon on static d--dimensional de Sitter space and thence finding the entropy by thermodynamics. Numeric evaluation yields the known answer i.e. (minus) the conformal anomaly on the d-sphere. The de Sitter quantities are obtained by conformal transformation of the Rindler ones, themselves obtained, for convenience, from those around a cosmic string. The expressions are given in terms of generalised Bernoulli polynomials for which an identity is derived. The arising spherical conformal anomaly is discussed and a formula is given for it for Branson's higher GJMS Laplacian, P_2k, as an oscillating polynomial in the level, k.

title: Notes on black-hole evaporation
abstract: This paper examines various aspects of black-hole evaporation. A two-dimensional model is investigated where it is shown that using fermion-boson cancellation on the stress-energy tensor reduces the energy outflow to zero, while other noncovariant techniques give the Hawking result. A technique for replacing the collapse by boundary conditions on the past horizon is developed which retains the essential features of the collapse while eliminating some of the difficulties. This set of boundary conditions is also suggested as the most natural set for a preexistent black hole. The behavior of particle detectors under acceleration is investigated where it is shown that an accelerated detector even in flat spacetime will detect particles in the vacuum. The similarity of this case with the behavior of a detector near the black hole is brought out, and it is shown that a geodesic detector near the horizon will not see the Hawking flux of particles. Finally, the work of Berger, Chitre, Nutku, and Moncrief on scalar geons is corrected, and the spherically symmetric coupled scalar-gravitation Hamiltonian is presented in the hope that someone can apply it to the problem of black-hole evaporation.

title: ON THE DUALITY CONDITION FOR QUANTUM FIELDS
abstract: A general quantum field theory is considered in which the fields are assumed to be operator‐valued tempered distributions. The system of fields may include any number of boson fields and fermion fields. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the fields. Extensions of the algebras of field operators are discussed with reference to the duality conditions. Local internal symmetries are discussed, and it is shown that these commute with the Poincare group and with the TCP transformation.

title: FIELD THEORIES ON CONFORMALLY RELATED SPACE-TIMES: SOME GLOBAL CONSIDERATIONS
abstract: The nature of the vacua appearing in the relation between the vacuum expectation value of stress tensors in conformally flat spaces is clarified. The simple but essential point is that the relevant spaces should have conformally related global Cauchy surfaces. Some commonly occurring conformally flat space-times are divided into two families according to whether they are conformally equivalent to Minkowski space or to the Rindler wedge. Expressions, some new, are obtained for the vacuum expectation value of the stress tensor for a number of illustrative cases. It is noted that thermalization relates the Green's functions of these two families.

title: Modular structure of the local algebras associated with the free massless scalar field theory
abstract: The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII1 factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.

title: AdS/CFT Duals of Topological Black Holes and the Entropy of Zero-Energy States
abstract: The horizon of a static black hole in anti-de Sitter space can be spherical, planar, or hyperbolic. The microscopic dynamics of the first two classes of black holes have been extensively discussed recently within the context of the AdS/CFT correspondence. We argue that hyperbolic black holes introduce new and fruitful features in this respect, allowing for more detailed comparisons between the weak and strong coupling regimes. In particular, by focussing on the stress tensor and entropy of some particular states, we identify unexpected increases in the entropy of Super-Yang-Mills theory at strong coupling that are not accompanied by increases in the energy. We describe a highly degenerate state at zero temperature and zero energy density. We also find that the entanglement entropy across a Rindler horizon in exact AdS5 is larger than might have been expected from the dual SYM theory. Besides, we show that hyperbolic black holes can be described as thermal Rindler states of the dual conformal field theory in flat space.

title: Stress tensors and their trace anomalies in conformally flat space-time
abstract: Stress-energy tensors of massless fields propagating in curved space-times which are conformally flat are determined entirely from their trace anomalies. This calculation utilizes the integrability condition obeyed by stress tensors for arbitrary variations of the metric and sheds some light on the nature of the dimensionally continued renormalization method.

title: The Large-N Limit of Superconformal Field Theories and Supergravity
abstract: We show that the large-N limits of certainconformal field theories in various dimensions includein their Hilbert space a sector describing supergravityon the product of anti-de Sitter spacetimes, spheres, and other compact manifolds. This is shown bytaking some branes in the full M/string theory and thentaking a low-energy limit where the field theory on thebrane decouples from the bulk. We observe that, in this limit, we can still trust thenear-horizon geometry for large N. The enhancedsupersymmetries of the near-horizon geometry correspondto the extra supersymmetry generators present in thesuperconformal group (as opposed to just the super-Poincaregroup). The 't Hooft limit of 3 + 1 N = 4 super-Yang–Mills at the conformal pointis shown to contain strings: they are IIB strings. Weconjecture that compactifications of M/string theory on various anti-de Sitterspacetimes is dual to various conformal field theories.This leads to a new proposal for a definition ofM-theory which could be extended to include fivenoncompact dimensions.

title: Anti-de Sitter space and holography
abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of ${\cal N}=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the ${\cal N}=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

title: Quasilocal energy and conserved charges derived from the gravitational action.
abstract: The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on [sup 3][ital B], the history of the system's boundary. Energy surface density, momentum surface density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate [sup 3][ital B]. The integral of the energy surface density over such a two-surface [ital B] is the quasilocal energy associated with a spacelike three-surface [Sigma] whose orthogonal intersection with [sup 3][ital B] is the boundary [ital B]. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary [ital B] as a surface embedded in [Sigma]. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper-time translations on [sup 3][ital B] in the timelike direction orthogonal to [ital B]. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on [sup 3][ital B]. For spacetimes that are asymptoticallymore » flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt, Deser, and Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.« less

title: Entanglement entropy: Holography and renormalization group
abstract: In this review the entanglement and Renyi entropies in quantum field theory are described from different points of view, including the perturbative approach and holographic dualities. The applications of these results to constraining renormalization group flows are presented effectively and illustrated with a variety of examples.

title: Holographic c-theorems in arbitrary dimensions
abstract: We re-examine holographic versions of the c-theorem and entanglement entropy in the context of higher curvature gravity and the AdS/CFT correspondence. We select the gravity theories by tuning the gravitational couplings to eliminate non-unitary operators in the boundary theory and demonstrate that all of these theories obey a holographic c-theorem. In cases where the dual CFT is even-dimensional, we show that the quantity that flows is the central charge associated with the A-type trace anomaly. Here, unlike in conventional holographic constructions with Einstein gravity, we are able to distinguish this quantity from other central charges or the leading coefficient in the entropy density of a thermal bath. In general, we are also able to identify this quantity with the coefficient of a universal contribution to the entanglement entropy in a particular construction. Our results suggest that these coefficients appearing in entanglement entropy play the role of central charges in odd-dimensional CFT’s. We conjecture a new c-theorem on the space of odd-dimensional field theories, which extends Cardy’s proposal for even dimensions. Beyond holography, we were able to show that for any even-dimensional CFT, the universal coefficient appearing the entanglement entropy which we calculate is precisely the A-type central charge.

title: The F-Theorem and F-Maximization
abstract: This contribution contains a review of the role of the three-sphere free energy F in recent developments related to the F-theorem and F-maximization. The F-theorem states that for any Lorentz-invariant RG trajectory connecting a conformal field theory CFT_UV in the ultraviolet to a conformal field theory CFT_IR, the F-coefficient decreases: F_UV > F_IR. I provide many examples of CFTs where one can compute F, approximately or exactly, and discuss various checks of the F-theorem. F-maximization is the principle that in an N=2 SCFT, viewed as the deep IR limit of an RG trajectory preserving N=2 supersymmetry, the superconformal R-symmetry maximizes F within the set of all R-symmetries preserved by the RG trajectory. I review the derivation of this result and provide examples.

title: Quantum inequalities and ``quantum interest'' as eigenvalue problems
abstract: Quantum inequalities (QI's) provide lower bounds on the averaged energy density of a quantum field. We show how the QI's for massless scalar fields in even dimensional Minkowski space may be reformulated in terms of the positivity of a certain self-adjoint operator - a generalised Schroedinger operator with the energy density as the potential - and hence as an eigenvalue problem. We use this idea to verify that the energy density produced by a moving mirror in two dimensions is compatible with the QI's for a large class of mirror trajectories. In addition, we apply this viewpoint to the `quantum interest conjecture' of Ford and Roman, which asserts that the positive part of an energy density always overcompensates for any negative components. For various simple models in two and four dimensions we obtain the best possible bounds on the `quantum interest rate' and on the maximum delay between a negative pulse and a compensating positive pulse. Perhaps surprisingly, we find that - in four dimensions - it is impossible for a positive delta-function pulse of any magnitude to compensate for a negative delta-function pulse, no matter how close together they occur.

title: Holographic heat engines, entanglement entropy, and renormalization group flow
abstract: We explore a fruitful connection between the physics of conformal field theories (CFTs) in d-dimensional Minkowski spacetime and the extended gravitational thermodynamics of hyperbolic black holes in (d+1)-dimensional anti-de Sitter spacetime. The CFTs are reduced on a region bounded by a sphere. We show that Renormalization Group (RG) flows between CFTs are specific thermodynamic processes in the (p,V) plane, where the irreversibility of coarse-graining flows from the ultraviolet to the infrared corresponds to the Second Law of thermodynamics, preventing heat from flowing from low temperature to high. We observe that holographic heat engines using the black holes as a working substance correspond to specific combinations of CFT flows and deformations. We construct three special engines whose net heat and work can be described in terms of changes of entanglement entropy across the sphere. Engine efficiencies emerge as simple functions of the ratio of the number of degrees of freedom of two CFTs.

title: Black hole chemistry: thermodynamics with Lambda
abstract: We review recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemical enthalpy. This leads to an extended dictionary for black hole thermodynamic quantities, in particular a notion of thermodynamic volume emerges for a given black hole spacetime. This volume is conjectured to satisfy the reverse isoperimetric inequality—an inequality imposing a bound on the amount of entropy black hole can carry for a fixed thermodynamic volume. New thermodynamic phase transitions naturally emerge from these identifications. Namely, we show that black holes can be understood from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. We also review the recent attempts at extending the AdS/CFT dictionary in this setting, discuss the connections with horizon thermodynamics, applications to Lifshitz spacetimes, and outline possible future directions in this field.

title: Entanglement entropy for relevant and geometric perturbations
abstract: A bstractWe continue the study of entanglement entropy for a QFT through a perturbative expansion of the path integral definition of the reduced density matrix. The universal entanglement entropy for a CFT perturbed by a relevant operator is calculated to second order in the coupling. We also explore the geometric dependence of entanglement entropy for a deformed planar entangling surface, finding surprises at second order.

title: Shape dependence of entanglement entropy in conformal field theories
abstract: A bstractWe study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on ℝ1,d−1$$ {\mathrm{\mathbb{R}}}^{1,d-1} $$. We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient CT appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient σCT=π224$$ \frac{\sigma }{C_T}=\frac{\pi^2}{24} $$ in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.

title: Some properties of the Noether charge and a proposal for dynamical black hole entropy.
abstract: We consider a general, classical theory of gravity with arbitrary matter fields in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian, $\bL$. We first show that $\bL$ always can be written in a ``manifestly covariant" form. We then show that the symplectic potential current $(n-1)$-form, $\th$, and the symplectic current $(n-1)$-form, $\om$, for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current $(n-1)$-form, $\bJ$, and corresponding Noether charge $(n-2)$-form, $\bQ$. We derive a general ``decomposition formula" for $\bQ$. Using this formula for the Noether charge, we prove that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy, $S_{dyn}$, of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of $\bL$, $\th$, and $\bQ$. However, the issue of whether this dynamical entropy in general obeys a ``second law" of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors.

title: Holographic calculations of Rényi entropy
abstract: A bstractWe extend the approach of  to a new calculation of the Rényi entropy of a general CFT in d dimensions with a spherical entangling surface, in terms of certain thermal partition functions. We apply this approach to calculate the Rényi entropy in various holographic models. Our results indicate that in general, the Rényi entropy will be a complicated nonlinear function of the central charges and other parameters which characterize the CFT. We also exhibit the relation between this new thermal calculation and a conventional calculation of the Rényi entropy where a twist operator is inserted on the spherical entangling surface. The latter insight also allows us to calculate the scaling dimension of the twist operators in the holographic models.

title: Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity
abstract: It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.

title: Seeing a c-theorem with holography
abstract: Using the anti-de Sitter conformal theory correspondence, we examine holographic renormalization group flows in a framework where the bulk gravity contains higher curvature interactions. This holographic model allows us to distinguish the flow of the different central charges in dual theory. For example, in four dimensions, one finds that the flow of the central charge a is naturally monotonic but that of c is not. Our results agree with Cardy's proposal to extend Zamolodchikov's c-theorem to higher dimensions. We are also led to formulate a novel c-theorem for a universal coefficient appearing in the entanglement entropy of the fixed point conformal theories in any (including an odd) number of dimensions.

title: IASSNS-HEP-93 / 87 On Geometric Entropy
abstract: We show that a geometrical notion of entropy, definable in flat space, governs the first quantum correction to the Bekenstein-Hawking black hole entropy. We describe two methods for calculating this entropy – a straightforward Hamiltonian approach, and a less direct but more powerful Euclidean (heat kernel) method. The entropy diverges in quantum field theory in the absence of an ultraviolet cutoff. Various related finite quantities can be extracted with further work. We briefly discuss the corresponding question in string theory.

title: A Comment on Entropy and Area
abstract: For an arbitrary quantum field in flat space with a planar boundary, an entropy of entanglement, associated with correlations across the boundary, is present when the field is in its vacuum state. The vacuum state of the same quantum field appears thermal in Rindler space, with an associated thermal entropy. We show that the density matrices describing the two situations are identical, and therefore that the two entropies are equal. We comment on the generality and significance of this result, and make use of it in analyzing the area and cutoff dependence of the entropy. The equivalence of the density matrices leads us to speculate that a planar boundary in Minkowski space has a classical entropy given by the Bekenstein--Hawking formula.

title: Remarks on geometric entropy
abstract: The recently discussed notion of geometric entropy is shown to be related to earlier calculations of thermal effects in Rindler space. The evaluation is extended to de Sitter space and to a two-dimensional black hole.

title: The gravity dual of Rényi entropy
abstract: A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge-gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Rényi entropies. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Rényi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Rényi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.

title: Holographic charged Rényi entropies
abstract: A bstractWe construct a new class of entanglement measures by extending the usual definition of Rényi entropy to include a chemical potential. These charged Rényi entropies measure the degree of entanglement in different charge sectors of the theory and are given by Euclidean path integrals with the insertion of a Wilson line encircling the entangling surface. We compute these entropies for a spherical entangling surface in CFT’s with holographic duals, where they are related to entropies of charged black holes with hyperbolic horizons. We also compute charged Rényi entropies in free field theories.

title: Holographic phases of Rényi entropies
abstract: A bstractWe consider Rényi entropies $ {S_n}=\frac{1}{1-n } \log\;\mathrm{Tr}{\rho^n} $ of conformal field theories in flat space, with the entangling surface being a sphere. The AdS/CFT correspondence relates this Rényi entropy to that of a black hole with hyperbolic horizon; as the Rényi parameter n increases the temperature of the black hole decreases. If the CFT possesses a sufficiently low dimension scalar operator the black hole will be unstable to the development of hair. Thus, as n is varied, the Rényi entropies will exhibit a phase transition at a critical value of n. The location of the phase transition, along with the spectrum of the reduced density matrix ρ, depends on the dimension of the lowest dimension non-trivial scalar operator in the theory.

title: Stress tensor of massless conformal quantum fields in hyperbolic universes
abstract: The vacuum expectation value of the renormalized stress tensor for a conformally coupled massless scalar field propagating in an arbitrary hyperbolic Robertson-Walker spacetime is calculated. The vacuum state used is that obtained by conformal transformation to the static hyperbolic universe: The result differs from calculations performed using the vacuum defined by conformal transformation to Minkowski spacetime, and is also different from earlier results obtained using the conformally static vacuum which are shown to be incorrect.

title: Vacuum instability in the Einstein open metric
abstract: SummaryWe study phase transitions in the background of an open, static, Einstein universe. The one-loop effective potential is computed both at zero and at finite temperature. The interplay between critical temperature an curvature is exploited. We show that in such a geometry the spontaneously broken symmetric phase is favoured at high curvature.RiassuntoSi studiano transizioni di fase nello sfondo di un universo di Einstein statico ed aperto. Si calcola il potenziale effettivo ad un cappio sia a temperatura zero che a temperatura finita. Si ricava la relazione tra temperatura critica e curvatura. Si mostra come la fase a simmetria spontaneamente rotta in una tale geometria sia favorita per grandi curvature.РезюмеМы исследуем фазовые переходы в фоне открытой статической вселенной Эйнштейна. В случае нулевой и конечной температуры вычисляется однопетельный эффективный потенциал. Исследуется взаимосвязь между критической температурой и кривизной. Мы показываем, что в такой геометрии фаза спонтанно нарушенной симметрии является предпочтителюной при больших значениях кривизны.

title: Comments on conformal Killing vector fields and quantum field theory
abstract: We give a comprehensive analysis of those vacuums for flat and conformally flat space-times which can be defined by timelike, hypersurface-orthogonal, conformal Killing vector fields. We obtain formulas for the difference in stress-energy density between any two such states and display the correspondence with the renormalized stress tensors. A brief discussion is given of the relevance of these results to quantum-mechanical measurements made by noninertial observers moving through flat space.

title: A new vacuum state in de Sitter space
abstract: We present a new vacuum state which can be defined in a region of de Sitter space. This region corresponds to a K =−1 Robertson-Walker coordinatization of de Sitter space. The renormalized expectation value of the stress tensor for the conformally invariant massless scalar field in this vacuum is evaluated, and is found to differ from the corresponding result in the usual de Sitter-invariant vacuum by the absence of a pure radiation term. We also indicate that this can be accounted for by regarding the de Sitter-invariant vacuum as a thermal state relative to the new vacuum.

title: Thermal Stress Tensors in Static Einstein Spaces
abstract: The Bekenstein-Parker Gaussian path-integral approximation is used to evaluate the thermal propagator for a conformally invariant scalar field in an ultrastatic metric. If the ultrastatic metric is conformal to a static Einstein metric, the trace anomaly vanishes and the Gaussian approximation is especially good. One then gets the ordinary flat-space expressions for the renormalized mean squared field and stress-energy tensor in the ultrastatic metric. Explicit formulas for the changes in and resulting from a conformal transformation of an arbitrary metric are found and used to take the Gaussian approximations for these quantities in the ultrastatic metric over to the Einstein metric. The result for is exact for de Sitter space and agrees closely with the numerical calculations of Fawcett and Whiting in the Schwarzschild metric. The result for is exact in de Sitter space and the Nariai metric and is close to Candelas's values on the bifurcation two-sphere in the Schwarzschild metric. Thus one gets a good closed-form approximation for the energy density and stresses of a conformal scalar field in the Hartle-Hawking state everywhere outside a static black hole.

title: Thermal Wightman Functions and Renormalized Stress Tensors in the Rindler Wedge
abstract: The Wightman functions in the Rindler portion of Minkowski space-time are presented for any value of the temperature and for massless spin fields up to s = 1 and the renormalized stress tensor relative to Minkowski vacuum is discussed. A gauge ambiguity in the vector case is pointed out.

title: A Relationship between scalar Green functions on hyperbolic and Euclidean Rindler spaces
abstract: We derive a formula connecting in any dimension the Green function on the (D + 1)-dimensional Euclidean Rindler space and the one for a minimally coupled scalar field with a mass m in the D-dimensional hyperbolic space. The relation takes a simple form in the momentum space where the Green functions are equal at the momenta (p0, p) for Rindler and for hyperbolic space with a simple additive relation between the squares of the mass and the momenta. The formula has applications to finite temperature Green functions, Green functions on the cone and on the (compactified) Milne spacetime. Analytic continuations and interacting quantum fields are briefly discussed.

title: Noise kernels of stochastic gravity in conformally-flat spacetimes
abstract: The central object in the theory of semiclassical stochastic gravity is the noise kernel which is the symmetric two point correlation function of the stress-energy tensor. Using the corresponding Wightman functions in Minkowski, Einstein and open Einstein spaces, we construct the noise kernels of a conformally coupled scalar field in these spacetimes. From them we show that the noise kernels in conformally-flat spacetimes, including the Friedmann-Robertson-Walker universes, can be obtained in closed analytic forms by using a combination of conformal and coordindate transformations.

title: Rényi entropies for free field theories
abstract: A bstractRényi entropies Sq are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q ≥ 0. For (d + 1)-dimensional conformal field theories, the Rényi entropies across Sd−1 may be extracted from the thermal partition functions of these theories on either (d + 1)-dimensional de Sitter space or $ \mathbb{R} \times {\mathbb{H}^d} $, where $ {\mathbb{H}^d} $ is the d-dimensional hyperbolic space. These thermal partition functions can in turn be expressed as path integrals on branched coverings of the (d + 1)-dimensional sphere and S1×$ {\mathbb{H}^d} $, respectively. We calculate the Rényi entropies of free massless scalars and fermions in d = 2, and show how using zeta-function regularization one finds agreement between the calculations on the branched coverings of S3 and on S1 × $ {\mathbb{H}^2} $. Analogous calculations for massive free fields provide monotonic interpolating functions between the Rényi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Rényi entropy calculations in d > 2.

title: GROUP THEORY AND THE HYDROGEN ATOM. II.
abstract: The internal 0(4) symmetry group of the nonrelativistic hydrogen atom is discussed and used to relate the various approaches to the bound-state problems. A more general group 0(1,4) of transformations is shown to connect the various levels, which appear as basis vectors for a continuous set of unitary representations of this noncompact group.

title: Statistical entropy and AdS / CFT correspondence in BTZ black holes
abstract: We study AdS/CFT correspondence in the case of AdS3. We obtain the statistical entropy of the BTZ black hole in terms of the correct central charge and the conformal dimensions for the states corresponding to the BTZ black hole. We point out the difference between our method and the old fashioned approaches based on SL(2,R) Wess-Zumino-Witten model or Liouville theory.

title: Higher-derivative gravity, surface terms, and string theory.
abstract: We consider adding Euler densities to the Einstein action as new gravity interactions in higher-dimensional theories. When the appropriate surface terms are included, the surface geometry is sufficient data for the boundary-value problem associated with the new Lagrangians. We also discuss the relevance of such interactions in the context of string theory.

title: Five-dimensional rotating black holes in Einstein–Gauss–Bonnet theory
abstract: We present arguments for the existence of five-dimensional rotating black holes with equal magnitude angular momenta in Einstein–Gauss–Bonnet theory with negative cosmological constant. These solutions posses a regular horizon of spherical topology and approach asymptotically an anti-de Sitter spacetime background. We discuss the general properties of these solutions and, using an adapted counterterm prescription, we compute their entropy and conserved charges.