Testing gauge/gravity duality on a quantum black hole Juan Maldacena Science 344, 806 (2014); DOI: 10.1126/science.1254597
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INSIGHTS | P E R S P E C T I V E S
gravity corrections are small. The GNewton expansion in the gravity theory (which is the standard semiclassical expansion in gravity) corresponds to the 1/N 2 expansion of the collection of interacting oscillators. The bulk theory—a 10-dimensional string theory—can in principle be quantized by using the rules of string theory to calculate the quantum gravity corrections. When the gravity theory is well approximated by an Einstein-type gravity theory (as opposed to a more complicated string theory), then the quantum mechanical system of oscillators is strongly coupled. Under these conditions, the anharmonic terms are very important and cannot be treated as small perturbations. The system of oscillators is providing a so-called “holographic” description of the physics inside the 10-dimensional spacetime. The quantum mechanical system at finite temperature has a corresponding gravity description that involves a black hole. The entropy of the black hole can be computed with the area formula of Hawking and Bekenstein. In the bulk theory, the computation is relatively simple—it involves finding a black hole solution and computing the area of its horizon. In contrast, the corresponding computation for the strongly interacting oscillators is more difficult. However, it has been done numerically (5), and agreement was found for this leading contribution, which is the term in the entropy of order N 2. Hanada et al. now go further and compute the order unity (N 0) term that corresponds to a quantum gravity correction in the bulk theory. In general, such quantum gravity corrections to the entropy also involve the entropy of Hawking radiation (see the figure). However, in this case there is a numerically
Testing gauge/gravity duality on a quantum black hole A numerical test shows that string theory can provide a self-consistent quantization of gravity By Juan Maldacena
n general relativity, gravity is formulated as a classical physics theory, and formulating its fully quantum version is a great challenge. Gauge/gravity duality conjectures that certain special quantum theories (typically gauge theories) are equivalent to theories of gravity in an emergent spacetime that has more dimensions than the ones appearing in the original quantum theory, and thereby provides an important bridge between quantum theory and gravity. A simple example involves a matrix quantum mechanics theory (1, 2) of interacting fundamental particles that gives rise to gravity in a 10-dimensional curved spacetime (3). Black holes in this spacetime are described by the quantum system at finite temperature. Testing the conjectured duality is difficult because the quantum system is very strongly coupled— it has no easy mathematical solution— when the standard Einstein-like gravity description is applicable. On page 882 of this issue, Hanada et al. (4) test this relaSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA. E-mail: [email protected]
tion by numerically calculating the entropy of a black hole in the gauge theory. They also calculate this entropy with a gravity theory that includes the first quantum gravity correction. The agreement between the two computations is evidence for the validity of the gauge/gravity duality, as well as for the internal consistency of string theory as a quantum theory of gravity. The quantum mechanical system used by Hanada et al. is conceptually simple. It can be viewed as a set of harmonic oscillators with additional anharmonic terms that are cubic or quartic in the coordinates of the harmonic oscillators. These oscillators are arranged into N × N matrices and have interactions that are invariant under the special unitary group of order N [SU(N)] symmetry. When N is very large, such theories are expected to have a convergent expansion in N. For example, their thermal free energies have a leading expansion term proportional to N 2 and subleading terms that are suppressed by powers of 1/N 2. The alternative description in terms of gravity theory (3) can be viewed as a kind of gravitational potential well with spherical symmetry. When N is large, the gravitational theory is weakly coupled and quantum
B + – –
t en Ev
on riz ho + Black hole
Not completely black holes. (A) The gravity pull of a black hole is so strong that no particles—not even photons—escape within the event horizon. (B) However, Hawking showed that the production of virtual particles in the vicinity of the event horizon, as demanded by quantum field theory, leads to radiation if one particle escapes and the other enters the black hole. Hanada et al. calculated quantum corrections to the entropy of certain black holes at finite temperature.
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PHOTO: (PANEL A) NASA/ESA/XMM-NEWTON/SCIENCE SOURCE
more important correction that is related to the potential divergences in quantum gravity. In the 10-dimensional gravity theory, the first quantum correction would be divergent. In string theory, this divergence is removed by the presence of strings, but a finite local and calculable counterterm remains. This term gives a correction to the black hole entropy, and it is this particular correction that was matched by the numerical computation. These quantum corrections had been previously computed in the context of extremal and near-extremal black holes [see, e.g., (6)]. These are charged black holes with the minimal or near-minimal mass, respectively, for a given charge. They are stable and do not emit Hawking radiation. In some cases, the previous computations even matched further subleading corrections. Hanada et al. can now match the corrections in a more generic finite-temperature situation. This calculation numerically tests quantum gravity in a context where string theory is crucial for obtaining the answer. In the usual quantization of gravity as an effective field theory, the correction that was considered would be proportional to an uncalculable counterterm. However, string theory provides a precise value for this counterterm (7) and is the one that reproduces a numerical computation with the interacting quantum oscillators. In the near future, similar methods could also match the term that comes from the entropy of the Hawking radiation itself and eventually could match results for other observables in the thermal ensemble that probe more detailed aspects of the 10-dimensional geometry. This numerical test is further evidence of the internal consistency of string theory (i.e., that it does indeed provide a self-consistent quantization of gravity) and provides further evidence that the gauge/gravity duality is correct. Of course, the 10-dimensional space under consideration here is not the same as the four-dimensional region of the multiverse where we live. However, one could expect that such holographic descriptions might also be possible for a region like ours. ■
ILLUSTRATION: C. BICKEL/SCIENCE
1. J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995). 2. T. Banks, W. Fischler, S. H. Shenker, L. Susskind, Phys. Rev. D 55, 5112 (1997). 3. N. Itzhaki, J. M. Maldacena, J. Sonnenschein, S. Yankielowicz, Phys. Rev. D 58, 046004 (1998). 4. M. Hanada, Y. Hyakutake, G. Ishiki, J. Nishimura, Science 344, 882 (2014). 5. M. Hanada, Y. Hyakutake, J. Nishimura, S. Takeuchi, Phys. Rev. Lett. 102, 191602 (2009). 6. A. Sen, http://arxiv.org/abs/1402.0109 (2014). 7. M. B. Green, M. Gutperle, P. Vanhove, Phys. Lett. B 409, 177 (1997). 10.1126/science.1254597
Targeting the host immune response to fight infection Strategies to modify immune responses to infection can be found in our genome By J. Kenneth Baillie
very year, infectious diseases kill millions of people worldwide. A close look at the modes of death from infection reveals something surprising: Often death is not attributable to a direct effect of the pathogen or of any toxin it produces; rather, it is the consequence of a systemic inflammatory response in the host (1). Our own immune system destroys us. This concept was observed long ago, but
The core problem is finding the right element of an immune response to target with a drug. Even if we were to possess a full knowledge of every connection in an unimaginably vast network of interactions among cells, proteins, and nucleic acids that are required to generate an immune response, we would still be uncertain as to where to intervene to help patients survive infection. The good news is that evolution may have encoded within our genomes a shortcut to these targets.
“The core problem is finding the right element of an immune response to target with a drug … When there is a way to confer a resistant state on a susceptible individual, the results have been impressive.” efforts to find effective therapies that alter the host response to infection to promote survival have largely failed. Targeting the infectious agent is, at present, the only successful strategy, but the relentless emergence of antimicrobial resistance is an inevitable problem. Some pathogens, such as influenza virus, can evolve de novo resistance with terrifying speed (2). With hindsight, it is unsurprising that pharmacological interventions to alter the host response to infection have not been effective. Our immune system has evolved to fight a moving target. Whereas the job of the heart has changed little, and hemoglobin binds the same oxygen, and even the circuitry required to generate consciousness need not be different from that of our early ancestors, immunity must change rapidly, again and again, every time a new pathogen appears or an old pathogen mutates. By its very nature, the immune system is expected to be a mire of complexity, interdependence, and redundancy.
Susceptibility to infectious disease is one of the most strongly inherited of all common disease traits (3). Specific genetic variants confer susceptibility or resistance to infection. Knowledge of these variants presents us with a challenge: Here is the genetic code required for resistance; all we need to do is find a way to confer it upon susceptible individuals. Genetic susceptibility or resistance tells us the one thing we most want to know about any component of a complex system: What will be the effect of intervening here? When there is a way to confer a resistant state on a susceptible individual, the results have been impressive. For example, a patient with HIV became resistant to the virus after a bone marrow transplant with cells from a donor whose cells were resistant to the virus (4). In this case, the patient Roslin Institute, University of Edinburgh, Midlothian EH25 9RG, UK, and Intensive Care Unit, Royal Infrmary of Edinburgh, Edinburgh EH16 4SA, UK. E-mail: [email protected]
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