- Series: Cambridge Monographs on Mathematical Physics
- Paperback: 340 pages
- Publisher: Cambridge University Press (July 3, 2000)
- Language: English
- ISBN-10: 0521654750
- ISBN-13: 978-0521654753
- Product Dimensions: 7.5 x 0.8 x 9.2 inches
- Shipping Weight: 1.5 pounds (View shipping rates and policies)
- Average Customer Review: 1 customer review
- Amazon Best Sellers Rank: #3,456,260 in Books (See Top 100 in Books)
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Loops, Knots, Gauge Theories (Cambridge Monographs on Mathematical Physics)
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"...the book is useful for an immersion in the subject, researchers working in the field, no doubt, will find it pleasant because of the precise natural style of the authors, who are pioneers and leaders in the field." Hugo A. Morales, Mathematical Reviews
This text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the four fundamental forces. This text reviews loop representation theory, Maxwell theory, Yang-Mills theories, lattice techniques, knot and braid theory, and describes applications in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research.
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As is readily apparent in the forward to this book, the authors favor the first approach, believing that quantum gauge theories, of which quantum electrodynamics is a primary example, offer the best hope for guidance in constructing a viable quantum gravity. They emphasize though a very particular aspect of these theories, namely that the requirement for gauge invariance forces one to view the "Wilson loops" as being the entities of primary importance. But more importantly, the authors assert that that Wilson loops allow one to gain information in the non-perturbative realm of quantum field theory. Calculations in non-perturbative quantum field theory are notoriously difficult, even though some progress has been made in the area of lattice gauge theories, so any insight the authors can offer in this regard is of utmost importance. Hence this book should be viewed as a study of quantum observables on the loop space. The authors hope that these observables, called `Schwinger functions' in the perturbative realm, will along with the differential equations and boundary conditions that determine them, will give a viable theory of quantum gravity.
The differential geometry of gauge theories is usually done using the formalism of principal fiber bundles. Classical gauge fields are viewed as sections of these bundles, and the results of non-trivial field interactions are compared from point to point by the use of parallel transport along curves defined in the base spaces of these bundles. This comparison is done with a `connection' on the bundle, and for a closed curve the failure of an entity to return to its original value after traversing the curve is taken to be a sign of non-trivial interactions or "curvature". Principal fiber bundles of course have an associated Lie group and elements of this group act on objects to parallel transport them along the closed curves. These group elements are thus dependent on the curve, and are called `holonomies'.
This is the classical picture, but what happens to this scenario in the quantum realm, and in this realm is it plausible to view it as a theory of quantum gravity? The authors spend the first six chapters discussing the loop group, and its use in the quantization of classical electrodynamics and classical Yang-Mills theory, as compared with what is done in the usual Hamiltonian formalism. The `quantum loop representation' plays a central role in their exposition, which is motivated by essentially two different approaches, one of which is essentially a Fourier transform of wavefunctions of the connection, while the other involves the quantization of a non-canonical algebra. In both cases the quantization procedure involves coming to grips with a constrained system, which as is well known is very challenging and the loop representation cannot be expected to be a panacea in this regard.
The trick involves the identification of the physical states taking into account diffeomorphism invariance and the Hamiltonian constraint. The authors do this for pure quantum gravity (no matter fields) using the Ashtekar formalism and `point-splitting' methods, reinforcing the idea of course that one is not going to escape the need for regularization, as is the case for all successful quantum field theories so far. The inclusion of matter fields is done for (uncharged) Weyl fermions, with the geometric interpretation that the Weyl part of the Hamiltonian is a translation operator in much the same way as the Hamiltonian in the case of pure gravity. The authors believe take this to mean that the loop representation for quantum gravity predicts the Dirac equation for fermions, but unfortunately they do not elaborate on this in much detail at all.
If loop quantization is to bring about a "unified" field theory in some sense then it must be able to show how the loops from one theory can be combined with the loops from another. The authors do this for the case of general relativity and electrodynamics, wherein a loop representation is introduced that is based on a single loop that accounts for the information of these two interacting theories. As expected, this involves enlarging the symmetry group SU(2) to U(2). They show that the wavefunctions for the unified loop representation depend on two loops, but that there is no effective distinction between the two loops. Generalizations to the case of Yang-Mills + general relativity are alluded to in the text but not discussed in any depth.