SPECIAL SECTION: TWAS SCIENCE FRONTIERS String theory: a framework for quantum gravity and various applications Spenta R. Wadia International Center for Theoretical Sciences and Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India hole thermodynamics. The discovery of the AdS/CFT In this semi-technical review we discuss string theory (and all that goes by that name) as a framework for a correspondence, as a precise realization of the holo- quantum theory of gravity. This is a new paradigm in graphic principle of black hole physics, has shed new theoretical physics that goes beyond relativistic quan- light on the solution of large N gauge theories and other tum field theory. We provide concrete evidence for this field theories at strong coupling. Given the diversity of proposal. It leads to the resolution of the ultra-violet concepts and techniques, string theory has a healthy inter- catastrophe of Einstein’s theory of general relativity face with various branches of mathematics and statistical and an explanation of the Bekenstein–Hawking en- mechanics. More recently, there have appeared connec- tropy (of a class of black holes) in terms of Boltzmann’s tions with fluid mechanics and strongly coupled condensed formula for entropy in statistical mechanics. We dis- matter systems. cuss ‘the holographic principle’ and its precise and We begin with a brief review of the current theories of consequential formulation in the AdS/CFT correspon- physics and their limitations. dence of Maldacena. One consequence of this corres- pondence is the ability to do strong coupling calculations in SU(N) gauge theories in terms of semi- Quantum mechanics and general relativity classical gravity. In particular, we indicate a connec- tion between dissipative fluid dynamics and the dyna- Quantum mechanics mics of black hole horizons. We end with a discussion of elementary particle physics and cosmology in the framework of string theory. We do not cover all Quantum mechanics is the established framework to de- aspects of string theory and its applications to diverse scribe the world of molecules, atoms, nuclei and their areas of physics and mathematics, but follow a few constituents. Its validity has been tested to very short dis- –18 paths in a vast landscape of ideas. tances like 10 m in high-energy collision experiments at CERN and Fermi Lab. In quantum mechanics the scale of quantum effects is set by Planck’s constant = = 1.05 Keywords: Cosmology, elementary particles, quantum × 10–27 erg s, and a new mathematical formulation is re- gravity, string theory. quired. Position and momentum do not commute xp – px = i=, and the Heisenberg uncertainty relation ΔxΔp ≥ = Introduction implies a limit to which we can localize the position of a point particle. This fuzziness that characterizes quantum In this article, we discuss the need for a quantum theory mechanics resolves, e.g. the fundamental problem of the of gravity to address some of the important questions of stability of atoms. physics related to the very early universe and the physics of blackholes. We will posit the case for string theory as a framework to address these questions. The bonus of Relativistic quantum field theory string theory is that it has the tenets of a unified theory of The application of quantum mechanics to describe relati- all interactions, electro-magnetism, weak and strong in- vistic particles, requires the framework of quantum field teractions, and gravitation. Given this, string theory pro- theory (QFT), which is characterized by = and the speed vides a framework to address some fundamental issues in of light c. The successful theories of elementary particle cosmology and elementary particle physics. Examples are physics, viz. electro-weak theory and the theory of strong dark matter, supersymmetric particles, dark energy, unifi- interactions are formulated in this framework. The inherent cation for all interactions, etc. incompatibility between a ‘continuous field’ and the notion Perhaps the most important success of string theory, in of a quantum fluctuation at a space–time point is resolved recent times, is in providing a microscopic basis of black- in the framework of renormalization theory and the con- cept of the renormalization group that was developed by 1 e-mail: [email protected] Wilson . 1252 CURRENT SCIENCE, VOL. 95, NO. 9, 10 NOVEMBER 2008 SPECIAL SECTION: TWAS SCIENCE FRONTIERS General relativity of a theory of this epoch of space–time, it would be im- possible to understand in a fundamental way the evolu- General relativity was conceived by Einstein to resolve tion of the universe after the ‘Big Bang’. In fact the very an apparent contradiction between special relativity and notion of a ‘initial time’ may lose meaning. Perhaps a Newton’s theory of gravitation. In special relativity, in- new framework may provide a new language in terms of teractions take a finite time to propagate due to the finite- which we may address such questions about the very early ness of the speed of light. But in Newton’s theory the universe. gravitational interaction is instantaneous! The resolution in general relativity is that space–time is not static and re- Black hole information paradox: Quantized general sponds to matter by changing its geometry (the metric of relativity also runs into difficulties with quantum me- space–time) in accordance with Einstein’s equations. chanics in the description of phenomena in the vicinity of General relativity is a successful theory (by success we the horizon of a black hole. A black hole is formed when mean it is experimentally well-tested) for distances R and a large mass is packed in a small volume, characterized 2 2 masses M characterized by R 2 APl, where APl = 2GNM/c by the radius rh = 2GNM/c . If so, then even light cannot (GN is Newton’s constant) is the Planck length. This in- escape from its interior and hence the name black hole. cludes a large range of phenomena in relativistic astro- The surface with radius rh is called the horizon of a black physics. General relativity also provides the framework of hole and it divides the space–time into two distinct re- the standard inflationary model of cosmology within which gions. The horizon of the black hole is a one way gate: If one successfully interprets the cosmic microwave back- you check in you cannot get out! However, Hawking in ground (CMB) data. 1974 realized that in quantum mechanics black holes radi- Now let us indicate the problem one encounters when ate. He calculated the temperature of a black hole of mass 3 one tries to quantize general relativity. M: T = =c /8πGNM. Using the first law of thermodynam- ics and Bekenstein’s heuristic proposal that the entropy Divergent quantum theory: Just like in Maxwell’s the- of a black hole is proportional to the area of its horizon, ory, where electro-magnetic waves exist in the absence of he arrived at one of the most important facts of quantum sources, Einstein’s equations predict ‘gravity’ waves. The gravity. The entropy of a large black hole is given by ‘graviton’ is the analogue of the ‘photon’. The graviton can be considered as a ‘particle’ in the quantum theory A c3 S = h , (1) with mass M = 0 and spin S = 2. It is a fluctuation of the bh 4=G geometry around flat Minkowski space–time. N Given this, one does the obvious like in any quantum where A is the area of the horizon of the black hole. In field theory. One discusses emission and absorption proc- h general, the black hole is characterized by its mass, charge esses of gravitons. The non-linearity of Einstein’s theory and angular momentum, and the Bekenstein–Hawking implies that besides the emission and absorption of gravi- formula (1) is valid for all of them. Note that it involves tons by matter, gravitons can be emitted and absorbed by all the three fundamental constants =, c and GN. Defining gravitons. These processes are characterized by a dimen- 3 APl = =GN/c (Planck area) we can write it suggestively as sionless ‘coupling constant’: E = EPl, where E is the energy 5 1/2 19 of the process and EPl = (=c /GN) 10 GeV. As long as A we are discussing processes in which E/E 1, the ef- Pl Sbh = . (2) fects of quantum fluctuations are suppressed and negligi- 4APl ble. When E = EPl 2 1 gravity is strongly coupled and graviton fluctuations are large. It is basically this fact that A/APl represents the number of degrees of freedom of the –70 2 renders the standard quantum theory of the metric fluc- horizon. In 3 + 1 dim, APl ≈ 2.6 × 10 m . This formula tuations around a given classical space-time badly diver- is a benchmark for any theory of quantum gravity to re- gent and meaningless. Note that in discussing quantum produce. gravity effects, we introduced the Planck energy which The fact that the entropy is proportional to the area of involves all the three fundamental constants of nature: =, the horizon and not the volume it encloses, gives a clue c and GN. that even though the degrees of freedom of the black hole are apparently behind the horizon, they seem to leave an Big Bang singularity: If we consider the Friedmann– imprint (hologram) on the horizon. Robertson–Walker (FRW) expanding universe solution Once the black hole is formed it will emit Hawking ra- and extrapolate it backward in time, the universe would diation. It is here that we run into a problem with quan- be packed in a smaller and smaller volume in the far past. tum mechanics. The quantum states that make up the 2 Once again when we reach the Planck volume APl, we black hole cannot be reconstructed from the emitted ra- would expect a breakdown of the quantized general relati- diation, even in principle, within the framework of gen- vity, due to large uncontrolled fluctuations.