Using Papadodimas and Raju construction of operators describing the interior of a black hole, we present a general relation between partition functions of operators describing inside and outside the black hole horizon. In particular for an eternal black hole the partition function of the interior modes may be given in terms those partition functions associated with the modes of left and right exteriors. By making use of this relation we observe that setting a finite UV cutoff will enforce us to have a cutoff behind the horizon whose value is fixed by the UV cutoff. The resultant cutoff is in agreement with what obtained in the context of holographic complexity.
We argue that the complementarity picture, as interpreted as a reference frame change represented in quantum gravitational Hilbert space, does not suffer from the "firewall paradox" recently discussed by Almheiri, Marolf, Polchinski, and Sully. A quantum state described by a distant observer evolves unitarily, with the evolution law well approximated by semi-classical field equations in the region away from the (stretched) horizon. And yet, a classical infalling observer does not see a violation of the equivalence principle, and thus a firewall, at the horizon. The resolution of the paradox li…
An effective field theory for infalling observers in the vicinity of a quasi-static black hole is given in terms of a freely falling lattice discretization. The lattice model successfully reproduces the thermal spectrum of outgoing Hawking radiation, as was shown by Corley and Jacobson, but can also be used to model observations made by a typical low-energy observer who enters the black hole in free fall at a prescribed time. The explicit short distance cutoff ensures that, from the viewpoint of the infalling observer, any quantum information that entered the black hole more than a scrambling …
We find radiation in an infalling frame and present an explicit analytic evidence of the failure of no drama condition by showing that an infalling observer finds an infinite negative energy density at the event horizon. The negative and positive energy density regions are divided by the newly defined zero-energy curve. The evaporating black hole is surrounded by the negative energy which can also be observed in the infalling frame.
The central challenge in trying to resolve the firewall paradox is to identify excitations in the near-horizon zone of a black hole that can carry information without injuring a freely falling observer. By analyzing the problem from the point of view of a freely falling observer, I arrive at a simple proposal for the degrees of freedom that carry information out of the black hole. An infalling observer experiences the information-carrying modes as ingoing, negative energy excitations of the quantum fields. In these states, freely falling observers who fall in from infinity do not encounter a f…
The firewall paradox states that an observer falling into an old black hole must see a violation of unitarity, locality, or the equivalence principle. Motivated by this remarkable conflict, we analyze the causal structure of black hole spacetimes in order to determine whether all the necessary ingredients for the paradox fit within a single observer's causal patch. We particularly focus on the question of whether the interior partner modes of the outgoing Hawking quanta can, in principle, be measured by an infalling observer. Since the relevant modes are spread over the entire sphere, we answe…
We present concrete construction of interior operators for a black hole which is perturbed by an infalling observer. The construction is independent from the initial states of the black hole while dependent only on the quantum state of the infalling observer. The construction has a natural interpretation from the perspective of the boundary operator's growth, resulting from the collision between operators accounting for the infalling and outgoing modes. The interior partner modes are created once the infalling observer measures the outgoing mode, suggesting that the black hole interior is obse…
We investigate an important question of Hawking-like radiation as seen by an infalling observer during gravitational collapse. Using the functional Schrodinger formalism we are able to probe the time dependent regime which is out of the reach of the standard approximations like the Bogolyubov method. We calculate the occupation number of particles whose frequencies are measured in the proper time of an infalling observer in two crucially different space-time foliations: Schwarzschild and Eddington-Finkelstein. We demonstrate that the distribution in Schwarzschild reference frame is not quite t…
We describe the experience of an observer falling into a black hole using the AdS/CFT correspondence. In order to do this, we reconstruct the local bulk operators measured by the observer along his trajectory outside the black hole. We then extend our construction beyond the black hole horizon. We show that this is possible because of an effective doubling of the observables in the boundary theory, when it is in a pure state that is close to the thermal state. Our construction allows us to rephrase questions about information-loss and the structure of the metric at the horizon in terms of more…
We develop the representation of infalling observers and bulk fields in the CFT as a way to understand the black hole interior in AdS. We first discuss properties of CFT states which are dual to black holes. We then show that in the presence of a Killing horizon bulk fields can be decomposed into pieces we call ingoing and outgoing. The ingoing field admits a simple operator representation in the CFT, even inside a small black hole at late times, which leads to a simple CFT description of infalling geodesics. This means classical infalling observers will experience the classical geometry in th…
Recently we pointed out that the black hole interior operators can be reconstructed by using the Hayden-Preskill recovery protocols. Building on this observation, we propose a resolution of the firewall problem by presenting a state-independent reconstruction of interior operators. Our construction avoids the non-locality problem which plagued the "$A=R_{B}$" or "$\text{ER}=\text{EPR}$" proposals. We show that the gravitational backreaction by the infalling observer, who simply falls into a black hole, disentangles the outgoing mode from the early radiation. The infalling observer crosses the …
We build on the observation by Hawking, Perry and Strominger that a global black hole space-time supports a large number of soft hair degrees of freedom to shed new light on the firewall argument by Almheiri, Marolf, Polchinski, and Sully. We propose that the soft hair Goldstone mode is encoded in a classical transition function that connects the asymptotic and near horizon region. The entropy carried by the soft hair is part of the black hole entropy and encoded in the outside geometry. We argue that the infalling observer automatically measures the classical value of the soft mode before rea…
The postulates of black hole complementarity do not imply a firewall for infalling observers at a black hole horizon. The dynamics of the stretched horizon, that scrambles and re-emits information, determines whether infalling observers experience anything out of the ordinary when entering a large black hole. In particular, there is no firewall if the stretched horizon degrees of freedom retain information for a time of order the black hole scrambling time.
The black hole information problem has motivated many proposals for new physics. One idea, known as state-dependence, is that quantum mechanics must be generalized to describe the physics of black holes, and that fixed linear operators do not provide the fundamental description of experiences for infalling observers. Instead, such experiences are to be described by operators with an extra dependence on the global quantum state. We show that any implementation of this idea strong enough to remove firewalls from generic states requires massive violations of the Born rule. We also demonstrate a s…
We investigate the effect of gravitational back-reaction on the black hole evaporation process. The standard derivation of Hawking radiation is re-examined and extended by including gravitational interactions between the infalling matter and the outgoing radiation. We find that these interactions lead to substantial effects. In particular, as seen by an outside observer, they lead to a fast growing uncertainty in the position of the infalling matter as it approaches the horizon. We argue that this result supports the idea of black hole complementarity, which states that, in the description of …
We investigate the recent black hole firewall argument. For a black hole in a typical state we argue that unitarity requires every quantum of radiation leaving the black hole to carry information about the initial state. An information-free horizon is thus inconsistent with unitary at every step of the evaporation process (in particular both before and after Page time). The required horizon-scale structure is manifest in the fuzzball proposal which provides a mechanism for holding up the structure. In this context we want to address the experience of an infalling observer and discuss the recen…
String theory provides an example of the kind of apparent inconsistency that the {\it Principle of Black Hole Complementarity\/} deals with. To a freely infalling observer a string falling through a black hole horizon appears to be a Planck size object. To an outside observer the string and all the information it carries begin to spread as the string approaches the horizon. In a time of order the ``information retention time'' it fills the entire area of the horizon.
We comment on a recent paper of Almheiri, Marolf, Polchinski and Sully who argue against black hole complementarity based on the claim that an infalling observer 'burns' as he approaches the horizon. We show that in fact measurements made by an infalling observer outside the horizon are statistically identical for the cases of vacuum at the horizon and radiation emerging from a stretched horizon. This forces us to follow the dynamics all the way to the horizon, where we need to know the details of Planck scale physics. We note that in string theory the fuzzball structure of microstates does no…
We argue that the following three statements cannot all be true: (i) Hawking radiation is in a pure state, (ii) the information carried by the radiation is emitted from the region near the horizon, with low energy effective field theory valid beyond some microscopic distance from the horizon, and (iii) the infalling observer encounters nothing unusual at the horizon. Perhaps the most conservative resolution is that the infalling observer burns up at the horizon. Alternatives would seem to require novel dynamics that nevertheless cause notable violations of semiclassical physics at macroscopic …
The AMPS paradox challenges black hole complementarity by apparently constructing a way for an observer to bring information from the outside of the black hole into its interior if there is no drama at its horizon, making manifest a violation of monogamy of entanglement. We propose a new resolution to the paradox: this violation cannot be explicitly checked by an infalling observer in the finite proper time they have to live after crossing the horizon. Our resolution depends on a weak relaxation of the no-drama condition (we call it "little drama") which is the "complementarity dual" of scramb…
Trans-planckian frequencies can be mimicked outside a black-hole horizon as a tail of an exponentially large amplitude wave that is mostly hidden behind the horizon. The present proposal requires implementing a final state condition. This condition involves only frequencies below the cutoff scale. It may be interpreted as a condition on the singularity. Despite the introduction of the cutoff, the Hawking radiation is restored for static observers. Freely falling observers see empty space outside the horizon, but are "heated" as they cross the horizon.
The dynamics of a probe particle or wrapped brane moving in the two-dimensional Rindler space can be described by a time-dependent tachyon field theory. Using knowledge of tachyon condensation, we learn that the infalling brane gets thermalised and produces open string pairs at the Hagedorn temperature when entering into the near-horizon Rindler wedge. It is shown that the Hagedorn temperature of the infalling brane is equal to the Hawking temperature of the host black hole detected in the same time coordinate. The infalling brane will decay completely into closed strings, mainly massive modes…
Within the framework of black hole complementarity, a proposal is made for an approximate interior effective field theory description. For generic correlators of local operators on generic black hole states, it agrees with the exact exterior description in a region of overlapping validity, up to corrections that are too small to be measured by typical infalling observers.
A candidate parameterization is introduced, in an effective field theory framework, for the quantum information transfer from a black hole that is necessary to restore unitarity. This in particular allows description of the effects of this information transfer in the black hole atmosphere, for example seen by infalling observers. In the presence of such information transfer, it is shown that infalling observers need not experience untoward violence. Moreover, the presence of general moderate-frequency couplings to field modes with high angular momenta offers a mechanism to enhance information …
Microstate geometries in string theory replace the black-hole horizon with a smooth geometric "cap" at the horizon scale. In geometries constructed using superstratum technology, this cap has the somewhat surprising property that induces very large tidal deformations on infalling observers that are far away from it. We find that this large-distance amplification of the tidal effects is also present in horizonless microstate geometries constructed as bubbling solutions, but can be tamed by suitably arranging the bubbles to reduce the strength of some of the gravitational multipole moments. Howe…
We investigate the effects of quintessence dark energy on the shadows of black hole, surrounded by various profiles of accretions. For the thin disk accretion, the images of the black hole comprises the dark region and bright region, including direct emission, lensing rings and photon rings. Although their details depend on the form of the emission, generically, the direct emission plays a major role for the observed brightness of the black hole, while the lensing ring makes a small contribution and photon ring makes a negligible contribution. The existence of cosmological horizon also plays a…
In this note I argue that a version of complementarity is possible which evades the need for the "firewalls" recently proposed by Almheiri, Marolf, Polchinski, and Sully to burn up observers falling into black hole horizons. In particular I claim that it is consistent for an infalling observer to fall through an "old" black hole horizon without burning up, without this observer or any other seeing information loss or a violation of low energy effective field theory in an unexpected place. The reason that AMPS find the opposite conclusion is because they attempt to use low energy physics to tra…
String theory provides numerous examples of duality between gravitational theories and unitary gauge theories. To resolve the black hole information paradox in this setting, it is necessary to better understand how unitarity is implemented on the gravity side. We argue that unitarity is restored by nonlocal effects whose initial magnitude is suppressed by the exponential of the Bekenstein-Hawking entropy. Time-slicings for which effective field theory is valid are obtained by demanding the mutual back-reaction of quanta be small. The resulting bounds imply that nonlocal effects do not lead to …
To make predictions for an eternally inflating "multiverse", one must adopt a procedure for regulating its divergent spacetime volume. Recently, a new test of such spacetime measures has emerged: normal observers - who evolve in pocket universes cooling from hot big bang conditions - must not be vastly outnumbered by "Boltzmann brains" - freak observers that pop in and out of existence as a result of rare quantum fluctuations. If the Boltzmann brains prevail, then a randomly chosen observer would be overwhelmingly likely to be surrounded by an empty world, where all but vacuum energy has redsh…
The Karlhede invariant is formed from the contraction of the covariant derivative of the Riemann tensor. It is a coordinate invariant that vanishes at the Schwarzschild event horizon $r=2m$. The vanishing of the invariant allows an observer to construct a local measuring device and use it to detect an event horizon while falling into a black hole. Recent proposals postulate the existence of a "firewall" at the event horizon that may incinerate an infalling observer. These proposals face an apparent paradox if a freely falling observer detects nothing special in the vicinity of the horizon. The…
The future apparent horizon of a black hole develops large stress energy due to quantum effects, unless the outgoing modes are in a thermal density matrix at the local Hawking temperature. It is shown for generic pure states that the deviation from thermality is so small that an infalling observer will see no drama on their way to the stretched horizon, providing a derivation of black hole complementarity after the Page time. Atypical pure states, and atypical observers, may of course see surprises, but that is not surprising.
Almheiri, Marolf, Polchinski, and Sully argued that, for a consistent black hole evaporation process, the horizon of a sufficiently old black hole should be replaced by a "firewall" at which an infalling observer burns up, which obviously leads to the violation of the equivalence principle. We propose that once the infalling partner of an outgoing Hawking particle approaches a black hole singularity, it experiences decoherence and the loss of its entanglement with the outgoing Hawking particle. This implies we would no longer need firewalls to avoid the black hole information paradox.
If quantum mechanics governs nature, black holes must evolve unitarily, providing a powerful constraint on the dynamics of quantum gravity. Such evolution apparently must in particular be nonlocal, when described from the usual semiclassical geometric picture, in order to transfer quantum information into the outgoing state. While such transfer from a disintegrating black hole has the dangerous potential to be violent to generic infalling observers, this paper proposes the existence of a more innocuous form of information transfer, to relatively soft modes in the black hole atmosphere. Simplif…
An accelerating observer sees a thermal bath of radiation at the Hawking temperature which is proportional to the acceleration. Also, in string theory there is a Hagedorn temperature beyond which one cannot go without an infinite amount of energy. Several authors have shown that in the context of Hawking radiation a limiting temperature for string theory leads to a limiting acceleration, which for a black hole implies a minimum distance from the horizon for an observer to remain stationary. We argue that this effectively introduces a cutoff in Rindler space or the Schwarzschild geometry inside…
We revisit the "state-dependence" of the map that we proposed recently between bulk operators in the interior of a large AdS black hole and operators in the boundary CFT. By refining recent versions of the information paradox, we show that this feature is necessary for the CFT to successfully describe local physics behind the horizon --- not only for single-sided black holes but even in the eternal black hole. We show that state-dependence is invisible to an infalling observer who cannot differentiate these operators from those of ordinary quantum effective field theory. Therefore the infallin…
We define bit models of evaporating black holes which incorporate the effects of replica wormholes. These enter as non-perturbative corrections to the gravitational inner product, arising from spacetime wormholes that connect replica black holes. The resulting models have exactly unitary evolution, predict measurements on Hawking radiation in accord with a Page curve for entropy, and are compatible with a smooth horizon for infalling observers.
We analyze string production in the background of a Schwarzschild black hole, after developing first quantized methods which capture string-theoretic nonadiabatic effects which can exceed naive extrapolations of effective field theory. Late-time infalling observers are strongly boosted in the near horizon region relative to early observers and formation matter. In the presence of large boosts in flat spacetime, known string and D-brane scattering processes exhibit enhanced string production, even for large impact parameter. This suggests the possibility that the nonadiabatic dynamics required …
It has been argued that when black holes are treated as quantum systems there are implications at the horizon and not just the singularity. Infalling observers will meet a firewall of high energy quanta. We argue that the question of whether an observer falling into a black hole experiences a smooth horizon or a firewall is identical to the question of whether Schrodinger's cat is either in a definite state, alive or dead, or in a superposition of the two. Since experience with real macro-systems indicate the former, the black hole state vector is seen to describe a set of decoherent alternati…
Recently, Almheiri et. al. argued, via a delicate thought experiment, that it is not consistent to simultaneosuly require that (a) Hawking radiation is pure, (b) effective field theory is valid outside a stretched horizon and (c) infalling observers encounter nothing unusual as they cross the horizon. These are the three fundamental assumptions underlying Black Hole Complementarity and the authors proposed that the most conservative resolution of the paradox is that (c) is false and the infalling observer burns up at the horizon (the horizon acts as a "firewall"). However, the firewall violate…
We investigate the gravitational collapse of a non-rotating $n$-dimensional BTZ black hole in AdS space in the context of both classical and quantum mechanics. This is done by first deriving the conserved mass of a "spherically" symmetric domain wall, which is taken as the classical Hamiltonian of the black hole. Upon deriving the conserved mass, we also point out that, for a "spherically" symmetric shell, there is an easy and straight-forward way of determining the conserved mass, which is related to the proper time derivative of the interior and exterior times. This method for determining th…
Recently, it has been argued that black hole complementarity is inconsistent by showing that, for an infalling observer, it would lead to the existence of a firewall near the black hole horizon, thereby violating the equivalence principle. If true, this would necessitate to give up on at least one of the postulates of black hole complementarity. In this comment I want to address an additional assumption that went into the conclusion, that the early outgoing Hawking radiation is entangled with the late radiation.
We discuss the black hole information problem, including the recent claim that unitarity requires a horizon firewall, emphasizing the role of decoherence and macroscopic superpositions. We consider the formation and evaporation of a large black hole as a quantum amplitude, and note that during intermediate stages (e.g., after the Page time), the amplitude is a superposition of macroscopically distinct (and decohered) spacetimes, with the black hole itself in different positions on different branches. Small but semiclassical observers (who are themselves part of the quantum amplitude) that fall…
We investigate gravitational collapse of a $(3+1)$-dimensional BTZ black string in AdS space in the context of both classical and quantum mechanics. This is done by first deriving the conserved mass per unit length of the cylindrically symmetric domain wall, which is taken as the classical Hamiltonian of the black string. In the quantum mechanical context, we take primary interest in the behavior of the collapse near the horizon and near the origin (classical singularity) from the point of view of an infalling observer. In the absence of radiation, quantum effects near the horizon do not chang…
An insightful viewpoint was proposed by Susskind about AMPS firewall: the region behind the firewall does not exist and the firewall is an extension of the singularity. In this work, we provided a possible picture of this idea by combining Newman's complex metric and Dvali-Gomez BEC black holes, which are Bose-Einstein condensates of N gravitons. The inner space behind the horizon is a realized imaginary space encrusted by the real space outside the horizon. In this way, the singularity extents to the horizon to make a firewall for the infalling observer. Some gravitons escape during the fluct…
We present a quantum theory of black hole (and other) horizons, in which the standard assumptions of complementarity are preserved without contradicting information theoretic considerations. After the scrambling time, the quantum mechanical structure of a black hole becomes that of an eternal black hole at the microscopic level. In particular, the stretched horizon degrees of freedom and the states entangled with them can be mapped into the near-horizon modes in the two exterior regions of an eternal black hole, whose mass is taken to be that of the evolving black hole at each moment. Salient …
We have investigated the thermodynamical properties of the universe with dark energy. It is demonstrated that in a universe with spacial curvature the natural choice for IR cutoff could be the apparent horizon radius. We shown that any interaction of pressureless dark matter with holographic dark energy, whose infrared cutoff is set by the apparent horizon radius, implying a constant effective equation of state of dark component in a universe. In addition we found that for the static observer in space, the comoving distance has a faster expansion than the apparent horizon radius with any spati…
Modes just outside the horizon of a typical old black hole are thermally entangled with distant Hawking radiation. This precludes their entangled purity with interior modes, leading to a firewall. Identifying the interior with the distant radiation ("A=R_B", "ER=EPR") can resolve the entanglement conflict. But the map must adjust for any interactions, or else the firewall will reappear if the Hawking radiation scatters off the CMB. With a self-correcting map, an infalling observer is unable to excite the vacuum near the horizon. This allows the horizon to be locally detected and so violates th…
We analytically study the linear response of a near-extremal Kerr black hole to external scalar, electromagnetic, and gravitational field perturbations. We show that the energy density, electromagnetic field strength, and tidal force experienced by infalling observers exhibit transient growth near the horizon. The growth lasts arbitrarily long in the extremal limit, reproducing the horizon instability of extremal Kerr. We explain these results in terms of near-horizon geometry and discuss potential astrophysical implications.
We consider the issue of information recovery for an object carrying energy and entropy into a black hole using the generalized entropy formalism, in the context of JT gravity where the backreaction problem can be solved exactly. We verify the main aspects of the Hayden-Preskill scenario but with some refinements. We show that the information is encoded in the Hawking radiation in a redundant way, as expected for a quantum error correcting code. We show how quantum extremal surfaces associated to information recovery have the form of a python's lunch and thereby show that the complexity of dec…
We present an explicit reconstruction of the interior of an AdS$_2$ black hole in Jackiw-Teitelboim gravity, that is entirely formulated in the dual SYK model and makes no direct reference to the gravitational bulk. We do this by introducing a probe "observer" in the right wormhole exterior and using the prescription of [arXiv:2009.04476] to transport SYK operators along the probe's infalling worldline and into the black hole interior, using an appropriate SYK modular Hamiltonian. Our SYK computation recovers the precise proper time at which signals sent from the left boundary are registered b…
Hawking effect was dug out of the gravitationally collapsed site forty years back when it was realised that quantum effects at the horizon could propagate outward to infinity giving rise to a thermal flux at late-times. However, the situation regarding non-asymptotic observers was never completely clear. Also, recently a debate has sprung in the community as to what would infalling observers perceive while crossing the horizon. We set out to settle this question and more at least semi-classically in the articles [arXiv:1304.2858] and [arXiv:1404.0684] with a fresh approach. We introduce a new …
Non-commutative geometries have been proposed as an approach to quantum gravity and have led to the construction of non-commutative black holes, whose interior singularities are purportedly eliminated due to quantum effects. Here we find evidence that these black holes are in fact unstable, with infalling matter near the Cauchy (inner) horizon being subject to an infinite blueshift of the type that has been repeatedly demonstrated for the Reissner-Nordstr\(\ddot{\text{o}}\)m black hole. This instability is present even when an ultraviolet cutoff (induced by anticipated non-commutative geometri…
We consider the information loss paradox at different angles, from the standard semi-classical approximation of General Relativity to the recently proposed scenarios of black holes evolution caused by effects of gravity quantization. Focusing on the Firewall proposal, we study the loop effects on the geometry and boundary conditions in black hole spacetimes and analyze the energy measured by an infalling observer near their horizons. As a result we obtain a trans-Planckian energy transition for the time-dependent black hole solution on the quantum-induced AdS background, the importance of whic…
We construct the first family of microstate geometries of near-extremal black holes, by placing metastable supertubes inside certain scaling supersymmetric smooth microstate geometries. These fuzzballs differ from the classical black hole solution macroscopically at the horizon scale, and for certain probes the fluctuations between various fuzzballs will be visible as thermal noise far away from the horizon. We discuss whether these fuzzballs appear to infalling observers as fuzzballs of fuzz or as fuzzballs of fire. The existence of these solutions suggests that the singularity of non-extrema…
It has been suggested [1] that the resolution of the information paradox for evaporating black holes is that the holes are surrounded by firewalls, bolts of outgoing radiation that would destroy any infalling observer. Such firewalls would break the CPT invariance of quantum gravity and seem to be ruled out on other grounds. A different resolution of the paradox is proposed, namely that gravitational collapse produces apparent horizons but no event horizons behind which information is lost. This proposal is supported by ADS-CFT and is the only resolution of the paradox compatible with CPT. The…
We follow the prevailing view that black holes do not destroy but rather process and release information in the form of Hawking radiation. By making certain conservative assumptions regarding the interior dynamics of the quantum system we suggest an outside observer could, in principle, recover the initial quantum state $\left|ψ\right\rangle $ before the black hole has evaporated half of its entropy. In the current framework the retention time is associated with the time scale for a local perturbation to become effectively undetectable (scrambling time). Also, we provide a scenario for storing…
Based on our recent work on the discretization of the radial AdS$_2$ geometry of extremal BH horizons,we present a toy model for the chaotic unitary evolution of infalling single particle wave packets. We construct explicitly the eigenstates and eigenvalues for the single particle dynamics for an observer falling into the BH horizon, with time evolution operator the quantum Arnol'd cat map (QACM). Using these results we investigate the validity of the eigenstate thermalization hypothesis (ETH), as well as that of the fast scrambling time bound (STB). We find that the QACM, while possessing a l…
Quantum fluctuations in the background geometry of a black hole are shown to affect the propagation of matter states falling into the black hole in a foliation that corresponds to observations purely outside the horizon. A state that starts as a Minkowski vacuum at past null infinity gets entangled with the gravity sector, so that close to the horizon it can be represented by a statistical ensemble of orthogonal states. We construct an operator connecting the different states and comment on the possible physical meaning of the above construction. The induced energy-momentum tensor of these sta…
