Replica Wormholes and Quantum Hair

Replica Wormholes and Quantum Hair 

https://arxiv.org/abs/2404.02331

Xavier Calmet, Stephen D.H. Hsu 

We discuss recent applications of Euclidean path integrals to the black hole information problem. In calculations with replica wormholes as the next-to-leading order correction to the Gibbons-Hawking saddlepoint, the radiation density matrix approaches a pure state at late times, following the Page curve. We compare unitary evaporation of black holes (in real time), mediated by calculable quantum hair effects, with the replica wormhole results. Both replica wormhole and quantum hair approaches imply that radiation states are macroscopic superpositions of spacetime backgrounds, invalidating firewall and monogamy of entanglement constructions. Importantly, identification of modes inside the horizon with radiation modes (i.e., large scale nonlocality across the horizon) is not required to provide a physical picture of unitary evaporation. Radiation modes can encode the interior information while still remaining independent degrees of freedom.

Wormholes dominate the Gibbons-Hawking saddlepoint of the Euclidean path integral after the Page time. This is because wormholes can connect the interiors of any two black holes i,j. At late times the number of such pairs grows as the dimensionality of the radiation Hilbert space squared. 

The wormholes connect BHs with macroscopically different recoil trajectories. This means the radiation approaches a pure state that is a macroscopic superposition - very similar to what our quantum hair expressions indicate.

Paradise Lost - Migdal, Polyakov, and Landau

This is a placeholder for a longer post I hope to expand on in the future, based on this essay: 

Paradise Lost - Alexander Migdal 

Migdal and Polyakov were two of the great Soviet physicists of their generation. Polyakov is on the upper left and Migdal the lower right.

Wikipedia: Migdal, Polyakov. 

The essay describes their education as young physicists. They were examined by Landau himself at age 15, and by age 19 had written a paper anticipating the Higgs Mechanism and the role of spontaneous symmetry breaking in gauge theory.

Migdal: Khalat was a genius of political intrigue. Being married into Inner Circle of the Soviet System (his wife Valya is the daughter of a legendary Revolution hero), he used all his connections and all the means to achieve his secret goal — assemble the best brains and let them Think Freely. 

On the surface, his pitch to the Party went as follows. “The West is attacking us for anti-Semitism. The best way to counter this slander is to create an Institute, where Jews are accepted, allowed to travel abroad and generally look happy. This can be a very small Institute, by standards of Atomic Project, it will have no secret military research, it will cost you very little, but it will help “Rasryadka” (Détente). These Jews will be so happy, they will tell all their Jewish friends in the West how well they live. And if they won’t –it is after all, us who decide which one goes abroad and which one stays home. They are smart kids, they will figure out which side of the toast is buttered.” 

As I put it, Khalat sold half of his soul to Devil and used the money to save another half. I truly respect him for that, now once I learned what it takes to create a startup and try to protect it against hostile world. 

As many crazy plans before it, this plan really worked. Best brains were assembled in Landau Institute, they were given a chance to happily solve problems without being forced to eat political shit like the whole country and – yes, they sometimes traveled abroad and made friends in the West. 

In a way the plan worked too well — we became so worldly and so free that we could no longer be controlled. And, needless to say, our friends in the West became closer to us that our curators in KGB.

I was in the 1990s generation of American physicists who had to contend on the job market with a stream of great theorists from the former Soviet Union. Both Migdal and Polyakov ended up at Princeton, and there were many others in their wake, closer to my age.

Quantum Hair During Gravitational Collapse (published version in Physical Review D)

https://journals.aps.org/prd/pdf/10.1103/PhysRevD.108.086012

This is a follow up to our earlier work on quantum gravitational corrections to the exterior graviton field of a compact object, also known as quantum hair. 

Here we follow the gravitational collapse of a dust ball and show that the quantum hair persists through the formation of a black hole horizon. The detailed calculations are possible due to an effective field theory formulation of quantum gravity in the long wavelength, low spacetime curvature limit.

Quantum Hair in Electrodynamics and Gravity (Eur. Phys. J. Plus)

This is the published version of the arxiv preprint previously discussed here.

We found it interesting that quantum hair can already be found using the familiar Euler-Heisenberg effective action, which results from integrating out the electron in QED. 

The paper also contains a general argument for why solutions to the semiclassical field equations resulting from the effective action (both in gravity and QED) carry more information about the state of the source than in classical physics. 

From the Conclusions: 

The quantum effective actions for both electrodynamics and gravity lead to field equations which couple a compact source (charge current or energy-momentum tensor) to external fields (electromagnetic or graviton field) in a manner which, generically, leads to quantum memory and quantum hair effects. External solutions of the field equations deviate, due to quantum corrections, from the familiar classical forms that satisfy the Gauss law. As a specific consequence, more information about the interior source configuration is encoded in the external field than in the classical theory. 

As specific applications, we considered semiclassical sources (large black hole, macroscopic charge distribution), which allowed us to solve the quantum corrected field equations by expanding around a classical solution. However, fully quantum statements regarding quantum hair are also possible, which do not, for example, require a semiclassical source. In [1–3] it was shown that the quantum state of a compact source (e.g., in an energy eigenstate or superposition thereof) determines certain aspects of the quantum state of its external field. In principle, measurements of the external fields can fully determine the interior state of a black hole.

Quantum Hair During Gravitational Collapse

This is a follow up to our earlier work on quantum gravitational corrections to the exterior graviton field of a compact object, also known as quantum hair. Here we follow the gravitational collapse of a dust ball and show that the quantum hair persists through the formation of a black hole horizon.

The detailed calculations are possible due to an effective field theory formulation of quantum gravity in the long wavelength, low spacetime curvature limit.
 

Quantum Hair During Gravitational Collapse 

https://arxiv.org/abs/2305.09466 

X. Calmet, R. Casadio, S. Hsu, F. Kuipers 

We consider quantum gravitational corrections to the Oppenheimer-Snyder metric describing time-dependent dust ball collapse. The interior metric also describes Friedmann-Lemaitre-Robertson-Walker cosmology and our results are interpreted in that context. The exterior corrections are an example of quantum hair, and are shown to persist throughout the collapse. Our results show the quantum hair survives throughout the horizon formation and that the internal state of the resulting black hole is accessible to outside observers.

 

 

Quantum gravitational corrections to particle creation by black holes (Physics Letters B)

This is the published version of our preprint https://arxiv.org/abs/2303.00310.

Quantum gravitational corrections to particle creation by black holes 

X. Calmet, S. Hsu, M. Sebastianutti 

We calculate quantum gravitational corrections to the amplitude for the emission of a Hawking particle by a black hole. We show explicitly how the amplitudes depend on quantum corrections to the exterior metric (quantum hair). This reveals the mechanism by which information escapes the black hole. The quantum state of the black hole is reflected in the quantum state of the exterior metric, which in turn influences the emission of Hawking quanta.

In earlier work we showed that the quantum state of a black hole is reflected in the quantum state of the exterior metric (outside the horizon). This violates classical intuitions, but can be shown explicitly using long wavelength effective field theory.

We calculated examples of small corrections to the external spacetime geometry which are sensitive to the internal BH state. In this paper we show that these corrections in turn affect Hawking radiation amplitudes. 

This means that the Hawking radiation state depends on the internal BH state. At the quantum level the hole is not black! We derive the results using both Hawking's original method and the tunneling method of Parikh and Wilczek.

 

While the focus of the new paper is explicit calculations, the big picture statement is:

The quantum state of the BH is reflected in the quantum state of its external gravitational field, which forms the background where the Hawking radiation originates. Radiation amplitudes are NOT independent of interior state.

Quantum Hair in Electrodynamics and Gravity (arXiv:2209.12798)

New paper!

Quantum Hair in Electrodynamics and Gravity 

Xavier Calmet, Stephen D. H. Hsu 

arXiv:2209.12798 

We demonstrate the existence of quantum hair in electrodynamics and gravity using effective action techniques. In the case of electrodynamics we use the Euler-Heisenberg effective action while in the case of quantum gravity we use the unique effective action. We give a general formulation of these effects which applies to both theories and discuss analogies and differences between them. Furthermore, we present a QED analog to black hole evaporation. Spontaneous pair production in the external field of a ball of charge is analogous to Hawking radiation from black holes. Assuming spherical symmetry, the Gauss law prevents the external field from depending on the density profile of the ball. Quantum corrections violate these expectations, showing that quantum radiation can encode classically forbidden information about the source.

We found it interesting that quantum hair can already be found using the familiar Euler-Heisenberg effective action, which results from integrating out the electron in QED.

The paper also contains a general argument for why solutions to the semiclassical field equations resulting from the effective action (both in gravity and QED) carry more information about the state of the source than in classical physics.

From the Conclusions:

The quantum effective actions for both electrodynamics and gravity lead to field equations which couple a compact source (charge current or energy-momentum tensor) to external fields (electromagnetic or graviton field) in a manner which, generically, leads to quantum memory and quantum hair effects. External solutions of the field equations deviate, due to quantum corrections, from the familiar classical forms that satisfy the Gauss law. As a specific consequence, more information about the interior source configuration is encoded in the external field than in the classical theory. 

As specific applications, we considered semiclassical sources (large black hole, macroscopic charge distribution), which allowed us to solve the quantum corrected field equations by expanding around a classical solution. However, fully quantum statements regarding quantum hair are also possible, which do not, for example, require a semiclassical source. In [1–3] it was shown that the quantum state of a compact source (e.g., in an energy eigenstate or superposition thereof) determines certain aspects of the quantum state of its external field. In principle, measurements of the external fields can fully determine the interior state of a black hole.

A Brief History of Hawking's Information Paradox (European Physics Letters)

This is a short review of our recent work on black hole information for European Physics Letters.

A Brief History of Hawking's Information Paradox 

European Physics Letters 139 (2022) 49001

Xavier Calmet and Stephen D. H. Hsu 

Abstract 

In this invited review, we describe Hawking's information paradox and a recently proposed resolution of it. Explicit calculations demonstrate the existence of quantum hair on black holes, meaning that the quantum state of the external graviton field depends on the internal state of the black hole. Simple quantum mechanics then implies that Hawking radiation amplitudes depend on the internal state, resulting in a pure final radiation state that preserves unitarity and, importantly, violates a factorization assumption which is central to the original paradox. Black hole information is encoded in entangled macroscopic superposition states of the radiation. 

DOI: 10.1209/0295-5075/ac81e8

arXiv version: https://arxiv.org/abs/2207.08671

From Conclusions:

... The radiation amplitudes computed by Hawking, which describe thermal radiation emitted from a black hole at temperature T, already describe a broad distribution of possible radiation types, spins, and momenta emitted at each stage. Thus, even in the semiclassical approximation there are many distinct patterns of radiation in (6). The set of possible final states is already complex even at leading order, resulting in very different coarse grained patterns of energy-momentum density. Small corrections to the amplitudes α(E, r) due to quantum hair do not qualitatively change this situation, but they are necessary to unitarize the evaporation and they determine the precise relations between components of the entangled state. 

Importantly, no special assumptions about the amplitudes α(E, r) need to be made to determine that the factorized form of the state (2) does not hold. Factorization is assumed in essentially every formulation of the information paradox, but in reality is violated because the external graviton state depends on the internal black hole state. 

Known quantum gravitational effects leading to quantum hair are extremely small and thus difficult to probe experimentally or detect via observations. We cannot prove that our solution to the information paradox is unique. However, the consequences of quantum hair lead, without any speculative theoretical assumptions, to plausible unitary evaporation of black holes. The properties of quantum hair and the evaporation amplitude (6) can be deduced using only long wavelength properties of quantum gravity – they do not rely on assumptions about Planck scale physics or a specific short distance completion. Therefore, Occam’s razor favors quantum hair.

Quantum Hair and Black Hole Information, University of Amsterdam, 17 Jun 2022

 

As promised, video from my talk in Amsterdam. 

Seminar at the Institute of Physics, University of Amsterdam, 17 Jun 2022. 

Title: Quantum Hair and Black Hole Information 

Abstract: I discuss recent results concerning the quantum state of the gravitational field of a compact matter source such as a black hole. These results demonstrate the existence of quantum hair, violating the classical No Hair Theorems. I then discuss how this quantum hair affects Hawking radiation, allowing unitary evaporation of black holes. Small corrections to leading order Hawking radiation amplitudes, with weak dependence on the external graviton state, are sufficient to produce a pure final radiation state. The radiation state violates the factorization assumption made in standard formulations of the information paradox. These conclusions are consequences of long wavelength properties of quantum gravity: no special assumptions are made concerning short distance (Planckian) physics. 

Slides

Institute of Physics, University of Amsterdam:

Has Hawking's Black Hole Information Paradox Been Resolved? (Video of MSU Theory Seminar 4/22/2022)

 

Theory seminar at Michigan State University April 22 2022. 

Title: Has Hawking's Black Hole Information Paradox Been Resolved? 

Abstract: In 1976 Stephen Hawking argued that black holes cause pure states to evolve into mixed states. Put another way, quantum information that falls into a black hole does not escape in the form of radiation. Rather, it vanishes completely from our universe, thereby violating a fundamental property of quantum mechanics called unitarity. I give a pedagogical introduction to this paradox, suitable for non-experts. Then I discuss recent results concerning the quantum state of the gravitational field of a compact matter source. These results demonstrate the existence of quantum hair, violating the classical No Hair Theorems. I then discuss how this quantum hair affects Hawking radiation, allowing unitary evaporation of black holes. 

Slides.

Black Hole Information and Quantum Hair in 10 minutes! (video)

 

This is a very nice 10 minute introduction to the black hole information paradox, and to our work on quantum hair. 

See earlier discussion Quantum Hair from Gravity (published version in Physical Review Letters). 

Parth G's video already has more than 10x as many views as my academic talk! Slides

Quantum Hair from Gravity (published version in Physical Review Letters)

This is the published version of our paper on Quantum Hair on black holes, in Physical Review Letters:

Quantum Hair from Gravity 

Xavier Calmet, Roberto Casadio, Stephen D. H. Hsu, and Folkert Kuipers 

Phys. Rev. Lett. 128, 111301 – Published 17 March 2022 

We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field. For a matter source in an energy eigenstate, the graviton state is determined at leading order by the energy eigenvalue. Insofar as there are no accidental energy degeneracies there is a one to one map between graviton states on the boundary of spacetime and the matter source states. Effective field theory allows us to compute a purely quantum gravitational effect which causes the subleading asymptotic behavior of the graviton state to depend on the internal structure of the source. This establishes the existence of ubiquitous quantum hair due to gravitational effects.

The paper establishes that the quantum state of the graviton field (equivalently, the spacetime metric) of a compact matter source depends on the quantum state of the source. This can be established without a short distance theory of quantum gravity -- i.e., near the Planck length. Our results are long wavelength effects and are insensitive to the details of short distance physics, such as whether gravitons are excitations of strings, or something else, at the most fundamental level.

Classical theorems in General Relativity indicate that black holes are nearly featureless -- only a few aspects of the hole, such as its total mass, charge, and angular momentum, are manifested in its asymptotic gravitational field. We show that this "no hair" property does not extend to the quantum realm. Indeed at the quantum level the situation is the opposite: the full quantum state of the compact object can be recovered from the asymptotic graviton state.

In this companion paper we show how these results resolve Hawking's black hole information paradox, which has been an open problem for 46 years.

Quantum hair and black hole information 

Physics Letters B Volume 827, 10 April 2022, 136995 

Xavier Calmet and Stephen D.H. Hsu 

It has been shown that the quantum state of the graviton field outside a black hole horizon carries information about the internal state of the hole. We explain how this allows unitary evaporation: the final radiation state is a complex superposition which depends linearly on the initial black hole state. Under time reversal, the radiation state evolves back to the original black hole quantum state. Formulations of the information paradox on a fixed semiclassical geometry describe only a small subset of the evaporation Hilbert space, and do not exclude overall unitarity.

Note to experts: the companion paper explains why Mathur's Theorem (i.e., entanglement entropy must always increase by ~ln 2 with each emitted qubit) is evaded once one considers BH evolution in the full radiation Hilbert space. The radiation Hilbert space is much larger than the small subspace which remains after conditioning on any specific spacetime background or BH recoil trajectory. Even exponentially small entanglement between different radiation states (mediated by quantum hair) can unitarize the evaporation process.

This is also explained in detail in the talk video and slides linked below.

Press coverage:

BBC

Guardian

Independent

Earlier discussion, with more background on the Hawking paradox. See especially the important work by Suvrat Raju and collaborators: 

Quantum Hair and Black Hole Information (December 2021) 

Has Hawking's Black Hole Information Paradox Been Resolved? (November 2021)

Video of seminar and slides.

Quantum Hair and Black Hole Information (Physics Letters B published version)

This is the published version of our recent arxiv preprint, previously discussed here.

Quantum hair and black hole information 

https://doi.org/10.1016/j.physletb.2022.136995 

Abstract 

It has been shown that the quantum state of the graviton field outside a black hole horizon carries information about the internal state of the hole. We explain how this allows unitary evaporation: the final radiation state is a complex superposition which depends linearly on the initial black hole state. Under time reversal, the radiation state evolves back to the original black hole quantum state. Formulations of the information paradox on a fixed semiclassical geometry describe only a small subset of the evaporation Hilbert space, and do not exclude overall unitarity.

The earlier paper, which established the existence of quantum hair, has been accepted by PRL and should also appear soon. 

Seminar video and slides. 

From the paper:

4. Conclusion 

Hawking's information paradox has been the focus of intense interest for almost 50 years. In his 1992 lecture on the subject, John Preskill wrote [5] 

I conclude that the information loss paradox may well presage a revolution in fundamental physics. 

The resolution described here is conservative: the quantum state of the exterior gravity field is determined by the interior black hole state, allowing the latter to influence Hawking radiation produced at the horizon. Two distinct quantum states of the black hole may produce the same semiclassical external geometry, but the graviton states differ at the quantum level. The relationship between interior and exterior quantum states is not governed by classical no-hair theorems. Indeed, it has gradually been appreciated that gravity itself prevents the localization of quantum information [4], [9], [10], [11], [21], [22], [23], even behind a horizon. We stress that all formulations of the paradox require a degree of factorization between the black hole internal state and the radiation (see, e.g., (6)), which is clearly not true of our equation (4). 

Certain aspects of our expressions (2)-(4) are very clear: the black hole information is spread over many branches of the final radiation state, and macroscopic superpositions of different spacetime geometries play a role in the evaporation. Some of the difficulty in resolving the paradox may originate from a reluctance to accept these aspects of quantum dynamics.

Black Hole Information and Quantum Hair: seminar video and slides

 

This is video of a seminar I gave at the University of Sussex. Slides.

Has Hawking's Black Hole Information Paradox Been Resolved? Quantum Hair and Black Hole Information 

Abstract: In 1976 Stephen Hawking argued that black holes cause pure states to evolve into mixed states. Put another way, quantum information that falls into a black hole does not escape in the form of radiation. Rather, it vanishes completely from our universe, thereby violating a fundamental property of quantum mechanics called unitarity. I give a pedagogical introduction to this paradox, suitable for non-experts. Then I discuss recent results concerning the quantum state of the gravitational field of a compact matter source. These results demonstrate the existence of quantum hair, violating the classical No Hair Theorems. I then discuss how this quantum hair affects Hawking radiation, allowing unitary evaporation of black holes.

See this related blog post: Has Hawking's Black Hole Information Paradox Been Resolved?

In the talk I mention an introductory colloquium on the history of black holes and the connection to entropy and information. See slides.

Quantum Hair and Black Hole Information

Our follow up paper on quantum hair is now on arXiv:

Quantum Hair and Black Hole Information 

https://arxiv.org/abs/2112.05171 

Xavier Calmet, Stephen D.H. Hsu 

It has been shown that the quantum state of the graviton field outside a black hole horizon carries information about the internal state of the hole. We explain how this allows unitary evaporation: the final radiation state is a complex superposition which depends linearly on the initial black hole state. Under time reversal, the radiation state evolves back to the original black hole quantum state. Formulations of the information paradox on a fixed semiclassical geometry describe only a small subset of the evaporation Hilbert space, and do not exclude overall unitarity.

This is the sequel to our earlier paper Quantum Hair from Gravity in which we first showed that the quantum state of the graviton field outside the black hole is determined by the quantum state of the interior.

Our results have important consequences for black hole information: they allow us to examine deviations from the semiclassical approximation used to calculate Hawking radiation and they show explicitly that the quantum spacetime of black hole evaporation is a complex superposition state.

The new paper describes Hawking evaporation of a black hole taking into account the quantum state of the exterior geometry.

After the first quantum hair paper appeared, I wrote a long post (November 14 2021) describing Hawking's black hole information paradox, which I excerpt from below. 

Has Hawking's Black Hole Information Paradox Been Resolved? 

In 1976 Stephen Hawking argued that black holes cause pure states to evolve into mixed states. Put another way, quantum information that falls into a black hole does not escape in the form of radiation. Rather, it vanishes completely from our universe, thereby violating a fundamental property of quantum mechanics called unitarity. 

These are bold statements, and they were not widely understood for decades. As a graduate student at Berkeley in the late 1980s, I tried to read Hawking’s papers on this subject, failed to understand them, and failed to find any postdocs or professors in the particle theory group who could explain them to me. 

As recounted in Lenny Susskind’s book The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics, he and Gerard ‘t Hooft began to appreciate the importance of black hole information in the early 1980s, mainly due to interactions with Hawking himself. In the subsequent decade they were among a very small number of theorists who worked seriously on the problem. I myself became interested in the topic after hearing a talk by John Preskill at Caltech around 1992:

Do Black Holes Destroy Information? 

https://arxiv.org/abs/hep-th/9209058 

John Preskill 

I review the information loss paradox that was first formulated by Hawking, and discuss possible ways of resolving it. All proposed solutions have serious drawbacks. I conclude that the information loss paradox may well presage a revolution in fundamental physics. 

Hawking’s arguments were based on the specific properties of black hole radiation (so-called Hawking radiation) that he himself had deduced. His calculations assumed a semiclassical spacetime background -- they did not treat spacetime itself in a quantum mechanical way, because this would require a theory of quantum gravity. 

Hawking’s formulation has been refined over several decades. 

Hawking (~1976): BH radiation, calculated in a semiclassical spacetime background, is thermal and is in a mixed state. It therefore cannot encode the pure state quantum information behind the horizon. 

No Cloning (~1990): There exist spacelike surfaces which intersect both the interior of the BH and the emitted Hawking radiation. The No Cloning theorem implies that the quantum state of the interior cannot be reproduced in the outgoing radiation. 

Entanglement Monogamy (~2010): Hawking modes are highly entangled with interior modes near the horizon, and therefore cannot purify the (late time) radiation state of an old black hole. 

However, reliance on a semiclassical spacetime background undermines all of these formulations of the BH information paradox, as I explain below. That is, there is in fact no satisfactory argument for the paradox. 

An argument for the information paradox must show that a BH evaporates into a mixed final state, even if the initial state was pure. However, the Hilbert space of the final states is extremely large: its dimensionality grows as the exponential of the BH surface area in Planck units. Furthermore the final state is a superposition of many possible quantum spacetimes and corresponding radiation states: it is described by a wavefunction of the form  ψ[g,M]  where g describes the spacetime geometry and M the radiation/matter fields.

It is easy to understand why the Hilbert space of [g,M] contains many possible spacetime geometries. The entire BH rest mass is eventually converted into radiation by the evaporation process. Fluctuations in the momenta of these radiation quanta can easily give the BH a center of mass velocity which varies over the long evaporation time. The final spread in location of the BH is of order the initial mass squared, so much larger than its Schwarzschild radius. Each radiation pattern corresponds to a complex recoil trajectory of the BH itself, and the resulting gravitational fields are macroscopically distinct spacetimes.

Restriction to a specific semiclassical background metric is a restriction to a very small subset X of the final state Hilbert space Y. Concentration of measure results show that for almost all pure states in a large Hilbert space Y, the density matrix 

 ρ(X) =  tr  ψ*ψ 

describing (small) region X will be exponentially close to thermal -- i.e., like the radiation found in Hawking's original calculation.

Analysis restricted to a specific spacetime background is only sensitive to the subset X of Hilbert space consistent with that semiclassical description. The analysis only probes the mixed state ρ(X) and not the (possibly) pure state which lives in the large Hilbert space Y. Thus even if the BH evaporation is entirely unitary, resulting in a pure final state ψ[g,M] in Y, it might appear to violate unitarity because the analysis is restricted to X and hence investigates the mixed state ρ(X). Entanglement between different X and X' -- equivalently, between different branches of the wavefunction ψ[g,M] -- has been neglected, although even exponentially small correlations between these branches may be sufficient to unitarize the result.

These and related issues are discussed in 

1. arXiv:0903.2258 Measurements meant to test BH unitarity must have sensitivity to detect multiple Everett branches 

2. arXiv:1302.0451  arXiv:1308.5686

BH evaporation leads to macroscopic superposition states; why this invalidates No Cloning and Entanglement Monogamy constructions, etc. Unitary evaporation does not imply unitarity on each semiclassical spacetime background.

See also Counting branches of the black hole wave function , Factorization of unitarity and black hole firewalls

3. arXiv:2011.11661 von Neumann Quantum Ergodic Theorem implies almost all systems evolve into macroscopic superposition states. Talk + slides.

When Hawking's paradox first received wide attention it was understood that the approximation of fixed spacetime background would receive quantum gravitational corrections, but it was assumed that these were small for most of the evaporation of a large BH. What was not appreciated (until the last decade or so) is that if spacetime geometry is treated quantum mechanically the Hilbert space within which the analysis must take place becomes much much larger and entanglement between X and X' supspaces which represent distinct geometries must be considered. In the "quantum hair" results it can be seen very explicitly that the evaporation process leads to entanglement between the radiation state, the background geometry, and the internal state of the hole. Within the large Hilbert space Y, exponentially small correlations (deviations from Hawking's original semiclassical approximation) can, at least in principle, unitarize BH evaporation.

In summary, my opinion for the past decade or so has been: theoretical arguments claiming to demonstrate that black holes cause pure states to evolve into mixed states have major flaws. 

This recent review article gives an excellent overview of the current situation: 

Lessons from the Information Paradox 

https://arxiv.org/abs/2012.05770 

Suvrat Raju 

Abstract: We review recent progress on the information paradox. We explain why exponentially small correlations in the radiation emitted by a black hole are sufficient to resolve the original paradox put forward by Hawking. We then describe a refinement of the paradox that makes essential reference to the black-hole interior. This analysis leads to a broadly-applicable physical principle: in a theory of quantum gravity, a copy of all the information on a Cauchy slice is also available near the boundary of the slice. This principle can be made precise and established — under weak assumptions, and using only low-energy techniques — in asymptotically global AdS and in four dimensional asymptotically flat spacetime. When applied to black holes, this principle tells us that the exterior of the black hole always retains a complete copy of the information in the interior. We show that accounting for this redundancy provides a resolution of the information paradox for evaporating black holes ...

Raju and collaborators have made important contributions demonstrating that in quantum gravity information is never localized -- the information on a specific Cauchy slice is recoverable in the asymptotic region near the boundary. [1] [2] [3]

However, despite the growing perception that the information paradox might be resolved, the mechanism by which quantum information inside the horizon is encoded in the outgoing Hawking radiation has yet to be understood. 

In a recent paper, my collaborators and I showed that the quantum state of the graviton field outside the horizon depends on the state of the interior. No-hair theorems in general relativity severely limit the information that can be encoded in the classical gravitational field of a black hole, but we show that this does not hold at the quantum level. 

Our result is directly connected to Raju et al.'s demonstration that the interior information is recoverable at the boundary: both originate, roughly speaking, from the Gauss Law constraint in quantization of gravity. It provides a mechanism ("quantum hair") by which the quantum information inside the hole can be encoded in ψ[g,M]. 

##########################

Below is a very nice talk by Raju given at the IAS workshop on Quantum Information and Spacetime just a week ago. Raju emphasizes that the external and internal BH states do not factorize, which is a key assumption in the information paradox. Quantum hair prevents factorization: it entangles the interior and exterior of the BH. 

Quantum Hair from Gravity

New paper!

Quantum Hair from Gravity 

https://arxiv.org/abs/2110.09386 

Xavier Calmet, Roberto Casadio, Stephen D. H. Hsu, and Folkert Kuipers 

We explore the relationship between the quantum state of a compact matter source and of its asymptotic graviton field. For a matter source in an energy eigenstate, the graviton state is determined at leading order by the energy eigenvalue. Insofar as there are no accidental energy degeneracies there is a one to one map between graviton states on the boundary of spacetime and the matter source states. A typical semiclassical matter source results in an entangled asymptotic graviton state. We exhibit a purely quantum gravitational effect which causes the subleading asymptotic behavior of the graviton state to depend on the internal structure of the source. These observations establish the existence of ubiquitous quantum hair due to gravitational effects.

From the introduction:

Classical no-hair theorems limit the information that can be obtained about the internal state of a black hole by outside observers [1]. External features (``hair'') of black hole solutions in general relativity are determined by specific conserved quantities such as mass, angular momentum, and charge. In this letter we investigate how the situation changes when both the matter source (black hole interior state) and the gravitational field itself are quantized. 

We begin by showing that the graviton state associated with an energy eigenstate source is determined, at leading order, by the energy eigenvalue of the source. These graviton states can be expressed as coherent states of non-propagating graviton modes, with explicit dependence on the source energy eigenvalue. Semiclassical matter sources (e.g., a star or black hole) are superpositions of energy eigenstates with support in some band of energies, and produce graviton states that are superpositions of the coherent states. ... We discuss implications for black hole information and holography in the conclusions.

General relativity relates the spacetime metric to the energy-momentum distribution of matter, but only applies when both the metric (equivalently, the gravitational field) and matter sources are semiclassical. A theory of quantum gravity is necessary to relate the quantum state of the gravitational field to the quantum state of the matter source. However, as we show in section 2 one can deduce this relationship either from a simple gedanken construction or from careful study of how the Gauss law affects quantization. It turns out the latter is common to both ordinary gauge theory (cf Coulomb potential) and gravity. 

Our results have important consequences for black hole information: they allow us to examine deviations from the semiclassical approximation used to calculate Hawking radiation and they show explicitly that the quantum spacetime of black hole evaporation is a complex superposition state.

See also 

Counting branches of the black hole wave function 

Factorization of unitarity and black hole firewalls

Feynman Lectures on the Strong Interactions (Jim Cline notes)

Professor James Cline (McGill University) recently posted a set of lecture notes from Feynman's last Caltech course, on quantum chromodynamics. Cline, then a graduate student, was one of the course TAs and the notes were meant to be assembled into a monograph. Thanks to Tim Raben for pointing these out to me.

The content seems a bit more elementary than in John Preskill's Ph230abc, a special topics course on QCD taught in 1983-4. I still consider John's notes to be one of the best overviews of nonperturbative aspects of QCD, a rather deep subject. However as Cline remarks there is unsurprisingly something special about the lectures: Feynman was an inspiring teacher, presenting everything in an incisive and fascinating way, that obviously had his own mark on it.

The material on QFT in non-integer spacetime dimensions is, as far as I know, original to Feynman. Dimensional regularization of gauge theory was popularized by 't Hooft and Veltman, but the analytic continuation to d = 4 - ε is specifc to the loop integrals (i.e., concrete mathematical expressions) that appear in perturbation theory. Here Feynman is, more ambitiously, exploring whether the quantum gauge theory itself can be meaningfully extended to a non-integer number of spacetime dimensions. 

Feynman Lectures on the Strong Interactions  

Richard P. Feynman, James M. Cline 

These twenty-two lectures, with exercises, comprise the extent of what was meant to be a full-year graduate-level course on the strong interactions and QCD, given at Caltech in 1987-88. The course was cut short by the illness that led to Feynman's death. Several of the lectures were finalized in collaboration with Feynman for an anticipated monograph based on the course. The others, while retaining Feynman's idiosyncrasies, are revised similarly to those he was able to check. His distinctive approach and manner of presentation are manifest throughout. Near the end he suggests a novel, nonperturbative formulation of quantum field theory in D dimensions. Supplementary material is provided in appendices and ancillary files, including verbatim transcriptions of three lectures and the corresponding audiotaped recordings.

The image below is from some of Feynman's handwritten notes (in this case, about the Gribov ambiguity in Fadeev-Popov gauge fixing) that Cline included in the manuscript. There are also links to audio from some of the lectures. As in some earlier notebooks, Feynman sometimes writes "guage" instead of gauge.

 

QED and QCD theta angles, asymptotic boundary conditions in gauge theory

Warning: this post is for specialists.

I had reason to look back at the paper below recently, and thought I would write a longer post on the subject as I regularly see searches on "QED theta angle" and similar in my traffic logs. These readers may be unsatisfied with the standard textbook treatment of this topic.

The conventional thinking is that because FF-dual in QED is a total derivative, doesn't affect the classical equations of motion, and isn't related to any topological vacuum structure, it can't have physical consequences.

In the paper below I constructed gauge configurations in (3+1) dimensions that connect an initial configuration (e.g., vacuum state A=0) to two different final configurations A1 and A2 (e.g., in the far future). A1 and A2 differ by a gauge transformation (i.e., represent the same physical electric and magnetic fields), but the two (3+1) interpolating configurations are not gauge equivalent. By construction, the value of \int FF-dual is not the same when evaluated on the two (3+1) configurations. Thus the two trajectories have a relative phase weighting in the path integral which depends on the value of theta. This suggests that theta can have physical (though perhaps small and non-perturbative) effects, so it is indeed a fundamental physical parameter of QED and of the Standard Model of particle physics.

http://arxiv.org/abs/1107.0756

Theta terms and asymptotic behavior of gauge potentials in (3+1) dimensions

We describe paths in the configuration space of (3+1) dimensional QED whose relative quantum phase (or relative phase in the functional integral) depends on the value of the theta angle. The final configurations on the two paths are related by a gauge transformation but differ in magnetic helicity or Chern-Simons number. Such configurations must exhibit gauge potentials that fall off no faster than 1/r in some region of finite solid angle, although they need not have net magnetic charge (i.e., are not magnetic monopoles). The relative phase is proportional to theta times the difference in Chern-Simons number. We briefly discuss some possible implications for QCD and the strong CP problem.

The question of whether physical observables can depend on the value of theta QED is somewhat esoteric. However, the analysis raises the issue of asymptotic boundary conditions in gauge theory. One typically expects that local properties of a quantum field theory are independent of the choice of boundary conditions when the size of the system is taken to infinity. But topological or total derivative terms such as FF-dual seem to defy this expectation.

The gauge potentials required for the construction described above must have  A ~ 1/r  behavior for some region of solid angle. In d=4 Euclidean space, the assumption is often made to allow only potentials with faster than 1/r falloff. In this way one obtains a topological classification of gauge configurations (i.e., in terms of instanton number). However, in Minkowski space (d=3+1) there exist classical solutions of non-Abelian gauge theory (e.g., SU(2); discovered by Luscher and Schecter) that exhibit 1/r falloff and are the analog of the U(1) configurations I described above. These L-S solutions have fractional topological charge.

In the presence of fractional topological charges the gauge theory partition function no longer appears periodic in theta. This may have consequences for the Strong CP problem in QCD, which are briefly discussed in the paper above.

Note Added: Writing this blog post was beneficial -- thinking through the topic again allowed me to formulate the conclusions more clearly than in the original paper. I've replaced it on arXiv with a new version containing the additional material below. The observation at the end is related to Elitzur's Theorem -- gauge-variant operators must have zero average in the path integral.

Farewell Freeman Dyson

He was 96 when he passed last Friday, one of the last giants who participated in the creation of Quantum Electrodynamics and modern quantum field theory.

He has appeared many times on this blog. Below are a few links.

The intestinal fortitude of Freeman Dyson: an account of his visit to the University of Oregon.

The evening began ominously. Dyson had a stomach bug -- he declined to eat anything at dinner, and made several emergency trips to the bathroom. After dinner he fell asleep on a couch in the physics building. Facing a packed auditorium, with people sitting in the aisles and filling an adjoining overflow room with video monitor, the other organizers and I decided that we'd offer Freeman the chance to call the whole thing off when we woke him up. Luckily for everyone, he felt much better after the nap, and was obviously energized by the large and enthusiastic crowd. After we finished the Q&A, he turned to me and said "Well, your questions cured my bug!"

Profile in The Atlantic (2010)

The prodigy in question, Freeman Dyson, now middle-aged, stared ahead, his incessant concentration on the road unbroken. ... I asked him whether as a boy he had speculated much about his gift. Had he asked himself why he had this special power? Why he was so bright?

Dyson is almost infallibly a modest and self-effacing man, but tonight his eyes were blank with fatigue, and his answer was uncharacteristic.

“That’s not how the question phrases itself,” he said. “The question is: why is everyone else so stupid?”

I want to emphasize that Dyson was a lovely and gentle person. The answer above is indeed uncharacterstic. But it is true...

Profile in NYTimes Magazine (2009). Below is a sample of work produced by Dyson between the ages of 5 and 9.

From Disturbing the Universe, one of my favorite scientific memoirs. It describes dramatic events of Dyson's early life: childhood in England, the war, QED, Feynman and Oppenheimer.

... In that spring of 1948 there was another memorable event. Hans [Bethe] received a small package from Japan containing the first two issues of a new physics journal. Progress of Theoretical Physics, published in Kyoto. The two issues were printed in English on brownish paper of poor quality. They contained a total of six short articles. The first article in issue No. 2 was called "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields," by S. Tomonaga of Tokyo University. Underneath it was a footnote saying, "Translated from the paper . . . (1943) appeared originally in Japanese." Hans gave me the article to read. It contained, set out simply and lucidly without any mathematical elaboration, the central idea of Julian Schwinger's theory. The implications of this were astonishing. Somehow or other, amid the ruin and turmoil of the war, totally isolated from the rest of the world, Tomonaga had maintained in Japan a school of research in theoretical physics that was in some respects ahead of anything existing anywhere else at that time. He had pushed on alone and laid the foundations of the new quantum electrodynamics, five years before Schwinger and without any help from the Columbia experiments. He had not, in 1943, completed the theory and developed it as a practical tool. To Schwinger rightly belongs the credit for making the theory into a coherent mathematical structure. But Tomonaga had taken the first essential Step. There he was, in the spring of 1948, sitting amid the ashes and rubble of Tokyo and sending us that pathetic little package. It came to us as a voice out of the deep.

A few weeks later, Oppy received a personal letter from Tomonaga describing the more recent work of the Japanese physicists. They had been moving ahead fast in the same direction as Schwinger. Regular communications were soon established. Oppy invited Tomonaga to visit Princeton, and a succession of Tomonaga's students later came to work with us at Princeton and at Cornell. When I met Tomonaga for the first time, a letter to my parents recorded my immediate impression of him "He is more able than either Schwinger or Feynman to talk about ideas other than his own. And he has enough of his own too. He is an exceptionally unselfish person." On his table among the physics journals was a copy of the New Testament.

Rest in Peace, Freeman Dyson.

The Quantum Theory of Fields

Excerpt from Sidney Coleman's Erice lectures. The period he describes just predates my entry into physics.

This was a great time to be a high-energy theorist, the period of the famous triumph of quantum field theory. And what a triumph it was, in the old sense of the word: a glorious victory parade, full of wonderful things brought back from far places to make the spectator gasp with awe and laugh with joy. I hope some of that awe and joy has been captured here.

Physics students learn quantum mechanics and special relativity as undergraduates, but typically do not encounter a synthesis of the two until graduate school, in a course on quantum field theory. Undergraduate quantum mechanics focuses on non-relativistic particles, moving at much less than the speed of light (e.g., the electrons in atomic systems or ordinary matter). Special relativity, as first encountered by students, is a modification of Newtonian (classical) mechanics, and ignores quantum effects.

In quantum field theory (QFT), the wave function of quantum mechanics Ψ(x) becomes a wave functional Ψ[ Φ(x) ], valued over field configurations Φ(x) which are themselves functions of spacetime coordinates. Individual particles are excitations ("quanta") of quantum fields. I think it is fair to say that almost no student really gets a deep understanding of quantum field theory when they take it for the first time. It is simply too complex to digest quickly. QFT introduces new intuitive pictures, novel calculational tricks, strange physical and mathematical constructs.

And how could it be otherwise? All of these tools are necessary to make sense of the generalization of ordinary quantum mechanics (of a finite number of degrees of freedom) to a physical system with an infinite number of degrees of freedom.

I first took quantum field theory (Physics 205) in my last year at Caltech, taught by Fredrik Zachariasen. Zachariasen used Bjorken and Drell I and II and Ramond as the main textbooks. He was what Russian theorists sometimes refer to as a "strong calculator" -- he would fill the blackboard with equations as fast as we could note them down. However, I would say his approach to the subject was rather old-fashioned by that time, and while I learned a good bit about the Dirac equation, spinors, how to compute Feynman diagrams, and even about path integrals, my overall understanding of the subject was still lacking. If I had been there the following year I would have enjoyed John Preskill's version of 205 (see below), but alas I was already in graduate school by then.

I remember that I also studied Feynman's short volume (in the Frontiers in Physics series; not to be confused with his later popular book) Quantum Electrodynamics. I was very confused at the time about the relationship between particles and fields and about so-called Second Quantization.  Also, what happened to the Schrodinger equation? At no point did Zachariasen (nor, I think, do Bjorken and Drell) clarify that while Dirac deduced his equation via relativistic generalization of Schrodinger's, the two are not on the same logical footing.

It was only some years later that I realized that Feynman himself had been confused about these things when he wrote his early papers on the subject. (Feynman, when someone explained a creation operator and Fock space to him: "How can you create an electron? It disagrees with conservation of charge!") Do Feynman diagrams describe spacetime trajectories of particles? Or are they simply graphical representations of terms in a perturbative expansion that happen to correspond, intuitively but not exactly, to physical processes?

As a first year graduate student at Berkeley I took Physics 230 from Stanley Mandelstam, a true master of the subject. This course was far more theoretical than the one I had taken the previous year. Amazingly, Stanley taught without notes. The only day he brought a single page of paper to class was when he covered the BPHZ proof of renormalizability. (Or was it the day he derived the beta function for non-Abelian gauge theories? I might be conflating two different instances.) His lectures followed no specific textbook, although the recommended one was probably Itzykson and Zuber.

My final student encounter with a QFT course was as the grader for Physics 230, taught by Martin Halpern. (I am sad to discover, in finding this link, that Marty passed away earlier this year.) Marty was a high strung chain smoker, and I recall many hours in his office going over solutions to his homework problems. He was especially on edge that fall because Vaughan Jones from the math department (who was about to share the Fields Medal with Ed Witten!) had decided to learn QFT and was sitting in on the class. As might be expected, the mathematician's insistence on clarity and precision slowed Marty down significantly. This wasn't Marty's fault -- QFT has not, even today, been placed on a completely rigorous footing (at least, not to the satisfaction of mathematicians), even though it is (in the form of Quantum Electrodynamics and the Standard Model) the most precisely tested theoretical construct in science.

This post is long enough. Perhaps I will revisit the topic in the future with a discussion of Sidney Coleman's lectures on QFT at Harvard, where I went after graduate school. It's nice to see that these lectures have been rendered into a book by his former students. For many years one could check out videotapes (Sony Betamax!) of his lectures from the physics library at Harvard. This made me think, even then, that the future of many professors might someday be as glorified teaching assistants, helping to explain and clarify recorded or streamed lectures by the true masters.

If I have kindled your interest in the subject, I recommend my friend Tony Zee's book: Quantum Field Theory in a Nutshell. Also, John Preskill's fantastic lecture notes, covering basic as well as advanced topics. It took me some time to learn to decipher his handwriting, but it was worth it!

Let me end by noting that the physics students who took these classes with me are quite a remarkable group. Among them are a number of well-known theoretical physicists, as well as the odd startup founder, AI researcher, or hedge fund billionaire. You could do worse in this life than get to know some students of quantum field theory :-)