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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Loop representation in gauge theories and quantum gravity ： ウィキペディア英語版 Loop representation in gauge theories and quantum gravity Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation, in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states (Gauss gauge invariant states). The idea is well known in the context of lattice Yang–Mills theory (see lattice gauge theory). Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations. The introduction by Ashtekar of a new set of variables (Ashtekar variables) cast general relativity in the same language as gauge theories and allowed one to apply loop techniques as a natural nonperturbative description of Einstein’s theory. In canonical quantum gravity the difficulties in using the continuous loop representation are cured by the spatial diffeomorphism invariance of general relativity. The loop representation also provides a natural solution of the spatial diffeomorphism constraint, making a connection between canonical quantum gravity and knot theory. Surprisingly there were a class of loop states that provided exact (if only formal) solutions to Ashtekar's original (ill-defined) Wheeler–DeWitt equation. Hence an infinite set of exact (if only formal) solutions had been identified for all the equations of canonical quantum general gravity in this representation! This generated a lot of interest in the approach and eventually led to loop quantum gravity (LQG). The loop representation has found application in mathematics. If topological quantum field theories are formulated in terms of loops, the resulting quantities should be what are known as knot invariants. Topological field theories only involve a finite number of degrees of freedom and so are exactly solvable. As a result, they provide concrete computable expressions that are invariants of knots. This was precisely the insight of Edward Witten〔E. Witten, Commun. Math. Phys 121, 351 (1989).〕 who noticed that computing loop dependent quantities in Chern–Simons and other three-dimensional topological quantum field theories one could come up with explicit, analytic expressions for knot invariants. For his work in this, in 1990 he was awarded the Fields Medal. He is the first and so far the only physicist to be awarded the Fields Medal, often viewed as the greatest honour in mathematics. == Gauge invariance of Maxwell's theory == The idea of gauge symmetries was introduced in Maxwell's theory. Maxwell's equations are $\backslash nabla\; \backslash cdot\; \backslash vec\; =\; \backslash qquad\; \backslash nabla\; \backslash times\; \backslash vec\; -\; \backslash epsilon\_0\; \backslash mu\_0\; =\; \backslash mu\_0\; \backslash vec\; \backslash qquad\; \backslash nabla\; \backslash times\; \backslash vec\; +\; =\; 0\; \backslash qquad\; \backslash nabla\; \backslash cdot\; \backslash vec\; =\; 0$ where $\backslash rho$ is the charge density and $\backslash vec$ the current density. The last two equations can be solved by writing fields in terms of a scalar potential, $\backslash phi$, and a vector potential, $\backslash vec$: $\backslash vec\; =\; -\; \backslash nabla\; \backslash phi\; -\; \backslash qquad\; \backslash vec\; =\; \backslash nabla\; \backslash times\; \backslash vec$. The potentials uniquely determine the fields, but the fields do not uniquely determine the potentials - we can make the changes: $\backslash phi\text{'}\; =\; \backslash phi\; +\; \backslash qquad\; \backslash vec\text{'}\; =\; \backslash vec\; -\; \backslash nabla\; \backslash Lambda$ without effecting the electric and magnetic fields, where $\backslash Lambda\; (\backslash vec,t)$ is an arbitrary function of space-time . These are called gauge transformations. There is an elegant relativistic notation: the gauge field is $A^\backslash mu\; =\; (\backslash phi\; ,\; \backslash vec)$ and the above gauge transformations read, $\text{'}\; =\; A^\backslash mu\; +\; \backslash partial^\backslash mu\; \backslash Lambda$. The so-called field strength tensor is introduced, $F^\; =\; \backslash partial^\backslash mu\; A^\backslash nu\; -\; \backslash partial^\backslash nu\; A^\backslash mu$ which is easily shown to be invariant under gauge transformations. In components, $F^\; =\; E^i\; ,\; \backslash qquad\; \backslash epsilon^\; F^\; =\; B^i$. Maxwell's source-free action is given by: $S\; =\; -\; \backslash int\; d^4x\; \backslash Big(\; F\_\; F^\; \backslash Big)$. The ability to varying the gauge potential at different points in space and time without changing the physics is called a local invariance. Electromagnetic theory possess the simplest kind of local gauge symmetry called $U\; (1)$. A theory that displays local gauge invariance is called a gauge theory. In order to formulate other gauge theories we turn the above reasoning inside out. This is the subject of the next section. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Loop representation in gauge theories and quantum gravity」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.