- Taschenbuch: 486 Seiten
- Verlag: Cambridge University Press (29. November 2007)
- Sprache: Englisch
- ISBN-10: 0521715962
- ISBN-13: 978-0521715966
- Größe und/oder Gewicht: 17 x 2,8 x 24,4 cm
- Durchschnittliche Kundenbewertung: 1 Kundenrezension
- Amazon Bestseller-Rang: Nr. 208.347 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
- Komplettes Inhaltsverzeichnis ansehen
Quantum Gravity (Cambridge Monographs on Mathematical Physics) (Englisch) Taschenbuch – 29. November 2007
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'In spite of its sociological success string theory is still far from a solution of the problem of quantum gravity which should be considered as wide open. The book of Carlo Rovelli provides the basis, both at the technical and the conceptual level, for research in this fundamental problem of physics. The basic issues are clearly and deeply analyzed without any dogmatic stand and with great freedom of thoughts resulting in an invaluable opportunity to learn and think for both mathematicians and physicists.' Alain Connes, Collège de France, I.H.E.S. and Vanderbilt University
'This is a truly great book! … a must for someone interested in quantum gravity.' Mathematical Reviews
'… written very carefully and can be recommended highly to anybody interested in the broad field and in conceptual issues of quantum gravity … experts … will find many new and clear notes of the author's view of loop quantum gravity as one of its founders … a non-expert reader, after working through the book, will be well prepared to read up further details in research or review articles …' General Relativity and Gravitation Journal
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Focusing on the conceptual and foundational issues raised by quantum gravity, in particular the nature of space and time, this 2004 book presents the loop and spinfoam approach. Aimed at graduate students entering the field, as well as established researchers, it will also be of interest to philosophers of science.Alle Produktbeschreibungen
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The initial five chapters of the book motivate the need for quantum gravity and also phrase the theories of general relativity and relativistic quantum physics in a language that will be used to formulate the theory of loop quantum gravity. General relativity (GR) is formulated as a dynamical system defined by the Hamilton-Jacobi equation for a functional defined on the space G of three-dimensional SU(2) connections. The equation is invariant under internal gauge transformations and 3D diffeomorphisms.
The quantization of GR is obtained in terms of complex-valued Schrodinger wave functionals on G. The derivatives of the Hamilton-Jacobi functional are replaced by derivative operators in order to obtain the quantum dynamics. The Hamilton-Jacobi equation then becomes the Wheeler-DeWitt equation governing the quantum dynamics of spacetime. The quantum (kinematical) state space is defined letting G be the space of smooth 3D real connections defined everywhere on a 3D surface S. The functionals are defined in terms of an ordered collection of smooth oriented paths L (essentially a lattice) on S, and are called "cylindrical functions" by the author. Scalar products are defined, which when completed gives the (kinematical) Hilbert space K. Lest the reader believe that this is nothing more than a quantum Yang-Mills theory on a lattice, the author is careful to note that a continuous theory over all possible lattices in S is being considered. The space K is nonseparable, but factoring out the diffeomorphisms gives a separable one. It has an orthonormal basis, and contains a subspace K0 of states invariant under local SU(2) gauge transformations. The ubiquitous spin network states form a orthonormal basis for K0. Again the author cautions against viewing this as ordinary lattice gauge theory, since diffeomorphism invariance makes it different from the latter. The spin networks are graphs L with links and nodes. The nodes are joined by links and the curves in L overlap only at the nodes. Each node has a multiplicity that measures the number of links going in and out of it. The author shows explicitly how to construct the spin network states, which are an orthornormal basis for K0.
So what about the observables of loop quantum gravity? The connection and its momentum are the field variables in the canonical theory, and are replaced by field operators. The momentum operator has a curious operation in this theory: the author describes it as "grasping" a path. The momentum operator though is not gauge invariant on K0, and so the author defines a new gauge-invariant (and self-adjoint) operator associated to S and has a straightforward operation on spin network states. This operator represents the physical area of S, and its spectrum, interestingly, can be interpreted as a quantized area. This result is related to the derivation of the entropy of black holes in the book, and is considered to be one of the significant results given by loop quantum gravity. A similar construction is done with the volume, giving a volume operator, which also has a discrete spectrum, but only has contributions from the nodes of a spin network state. Loop quantum gravity therefore truly gives a "quantized geometry." Each node of a spin network represents a quantum of volume, giving a collection of "chunks" separated from each other by surfaces, the areas of which are governed by the area operator. The graph L of the spin network determines the adjacency relation between these chunks, and is interpreted as the graph dual to a cellular decomposition of physical space. Spin network states therefore determine a discrete quantized three-dimensional metric.
The dynamics of loop quantum gravity requires the construction of the Hamiltonian operator. As in quantum field theories, this involves regularization, and the author shows how the Hamiltonian operator acts only on the nodes of the spin network, and gives a detailed discussion of the background independence of the theory. The latter is one of the most important features of loop quantum gravity, and is frequently advertised as one of its virtues. The author also discusses the extent to which the Hamiltonian operator is unique, outlining in the process several alternatives. When matter fields are considered, the author shows that the total Hamiltonian is finite, again pointing to the background-independence of loop quantum gravity. Loop quantum gravity reduces to classical general relativity as Planck's constant goes to zero, but the author lists many issues that have not been settled by this theory. One of these of course concerns the observable consequences, the lack of which it shares with string theory. Loop quantum gravity also does not attempt to unify the different interactions in nature in a single theory, as does string theory. But loop quantum gravity does give some interesting predictions, one of these being that the size of the universe is quantized. It also predicts an inflationary phase in the expansion of the early universe, as numerical solutions of the Wheeler-DeWitt equation indicate. By far the most interesting consequence of loop quantum gravity, is that it makes more reasonable the Bekenstein-Hawking interpretation of the entropy of a black hole. In fact the Bekenstein-Hawking entropy can be derived in loop quantum gravity, up to a factor called the Immirzi parameter. These are all impressive achievements, considering the status of quantum gravity now as compared to three decades ago.