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P H Y S I C S W O R L D N O V E M B E R 2 0 0 3 p h y s i c s w e b . o r g 1 G ENERAL relativity and quantum the- ory have profoundly changed our view of the world. Furthermore, both theo- ries have been verified to extraordinary accuracy in the last several decades. Loop quantum gravity takes this novel view of the world seriously, by incorpo- rating the notions of space and time from general relativity directly into quantum field theory. The theory that results is radically different from con- ventional quantum field theory. Not only does it provide a precise mathemat- ical picture of quantum space and time, but it also offers a solution to long-stand- ing problems such as the thermodynam- ics of black holes and the physics of the Big Bang. The most appealing aspect of loop quantum gravity is that it predicts that space is not infinitely divisible, but that it has a granular structure. The size of these elementary “quanta of space” can be computed explicitly within the the- ory, in an analogous way to the energy levels of the hydrogen atom. In the last 50 years or so, many approaches to con- structing a quantum theory of gravity have been explored, but only two have reached a full mathematical description of the quantum properties of the gravitational field: loop gravity and string theory. The last decade has seen major advances in both loop gravity and string theory,but it is important to stress that both theories harbour unresolved issues. More impor- tantly, neither of them has been tested experimentally. There is hope that direct experimental support might come soon, but for the moment either theory could be right, partially right or simply wrong.However,the fact that we have two well developed, tentative theories of quantum gravity is very encouraging. We are not completely in the dark, nor lost in a multitude of alternative theories, and quantum gravity offers a fascinating glimpse of the fundamental structure of nature. Space and quantum space Loop quantum gravity changes the way we think about the structure of space. To illustrate this, let me start by recalling some basic ideas about the notion of space and the way these were modified by general relativity. Space is commonly thought of as a fixed background that has a geometrical struc- ture – as a sort of “stage” on which mat- ter moves independently. This way of understanding space is not, however, as old as you might think; it was introduced by Isaac Newton in the 17th century. Indeed,the dominant view of space that was held from the time of Aristotle to that of Descartes was that there is no space without matter. Space was an abstraction of the fact that some parts of matter can be in touch with others. Newton introduced the idea of physi- cal space as an independent entity because he needed it for his dynamical theory. In order for his second law of motion to make any sense, acceleration must make sense. Newton assumed that there is a physical background space with respect to which acceleration is defined. The Newtonian picture of

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