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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 Ashtekar variables ： ウィキペディア英語版 Ashtekar variables In the ADM formulation of General relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, $q\_\; (x)$, on the spatial slice (the metric induced on the spatial slice by the spacetime metric and its conjugate momentum variable is related to the extrinsic curvature, $K^\; (x)$, (this tells us how the spatial slice curves with respect to spacetime and is a measure of how the induced metric evolves in time).〔''Gravitation'' by Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, published by W. H. Freeman and company. New York.〕 These are the metric canonical coordinates. In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar (new) variables to represent an unusual way of rewriting the metric canonical variables on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable.〔Ashtekar, A. (1986) ''Phys. Rev. Lett. 57, 2244''.〕 Ashtekar variables provide what is called the connection representation of canonical general relativity, which led to the loop representation of quantum general relativity〔Rovelli, C. and Smolin, L. ''Phys. Rev. Lett 61, 1155''〕 and in turn loop quantum gravity and quantum holonomy theory. Let us introduce a set of three vector fields $E^a\_i$, $i\; =\; 1,2,3$ that are orthogonal, that is, $\backslash delta\_\; =\; q\_\; E\_i^a\; E\_j^b$. The $E\_i^a$ are called a drei-bein or triad. There are now two different types of indices, "space" indices $a,b,c$ that behave like regular indices in a curved space, and "internal" indices $i,j,k$ which behave like indices of flat-space (the corresponding "metric" which raises and lowers internal indices is simply $\backslash delta\_$). Define the dual drei-bein $E^i\_a$ as $E^i\_a\; =\; q\_\; E^b\_i$. We then have the two orthogonality relationships $\backslash delta^\; =\; q^\; E^i\_a\; E^j\_b$ where $q^$ is the inverse matrix of the metric $q\_$ (this comes from substituting the formula for the dual drei-bein in terms of the drei-bein into $q^\; E^i\_a\; E^j\_b$ and using the orthogonality of the drei-beins). and $E\_i^a\; E^i\_b\; =\; \backslash delta\_b^a$ (this comes about from contracting $\backslash delta\_\; =\; q\_\; E\_j^b\; E\_i^a$ with $E^i\_c$ and using the linear independence of the $E\_a^j$). It is then easy to verify from the first orthogonality relation (employing $E\_i^a\; E^i\_b\; =\; \backslash delta\_b^a$) that $q^\; =\; \backslash sum\_^\; \backslash delta\_\; E\_i^a\; E\_j^b\; =\; \backslash sum\_^\; E\_i^a\; E\_i^b,$ we have obtained a formula for the inverse metric in terms of the drei-beins - the drei-beins may be thought of as the "square-root" of the metric (the physical meaning to this is that the metric $q^$, when written in terms of a basis $E\_i^a$, is locally flat). Actually what is really considered is $(\backslash mathrm\; (q))\; q^\; =\; \backslash sum\_^\; \backslash tilde\_i^a\; \backslash tilde\_i^b,$, which involves the densitized drei-bein $\backslash tilde\_i^a$ instead (densitized as $\backslash tilde\_i^a\; =\; \backslash sqrt\; E\_i^a$). One recovers from $\backslash tilde\_i^a$ the metric times a factor given by its determinant. It is clear that $\backslash tilde\_i^a$ and $E\_i^a$ contain the same information, just rearranged. Now the choice for $\backslash tilde\_i^a$ is not unique, and in fact one can perform a local in space rotation with respect to the internal indices $i$ without changing the (inverse) metric. This is the origin of the $SU\; (2)$ gauge invariance. Now if one is going to operate on objects that have internal indices one needs to introduce an appropriate derivative (covariant derivative), for example the covariant derivative for the object $V\_i^b$ will be $D\_a\; V\_i^b\; =\; \backslash partial\_a\; V\_i^b\; -\; \backslash Gamma\_^\; V\_j^b\; +\; \backslash Gamma^b\_\; V\_i^c$ where $\backslash Gamma^b\_$ is the usual Levi-Civita connection and $\backslash Gamma\_^$ is the so-called spin connection. Let us take the configuration variable to be $A\_a^i\; =\; \backslash Gamma\_a^i\; +\; \backslash beta\; K\_a^i$ where $\backslash Gamma\_a^i\; =\; \backslash Gamma\_\; \backslash epsilon^$ and $K\_a^i\; =\; K\_\; \backslash tilde^\; /\; \backslash sqrt$. The densitized drei-bein is the conjugate momentum variable of this three-dimensional SU(2) gauge field (or connection) $A^j\_b$, in that it satisfies the Poisson bracket relation $\backslash \; =\; 8\backslash pi\; G\_\}$. The densitized drei-bein can be used to re construct the metric as discussed above and the connection can be used to reconstruct the extrinsic curvature. Ashtekar variables correspond to the choice $\backslash beta\; =\; -i$ (the negative of the imaginary number), $A\_a^i$ is then called the chiral spin connection. The reason for this choice of spin connection was that Ashtekar could much simplify the most troublesome equation of canonical general relativity, namely the Hamiltonian constraint of LQG; this choice made its second, formidable, term vanish and the remaining term became polynomial in his new variables. This raised new hopes for the canonical quantum gravity programme.〔See the book ''Lectures on Non-Perturbative Canonical Gravity'' for more details on this and the subsequent development. First published in 1991. World Scientific Publishing Co. Pte. LtD.〕 However it did present certain difficulties. Although Ashtekar variables had the virtue of simplifying the Hamiltonian, it has the problem that the variables become complex.〔See part III chapter 5 of ''Gauge Fields, Knots and Gravity'', John Baez, Javier P. Muniain. First published 1994. World scientific Publishing Co. Pte. LtD.〕 When one quantizes the theory it is a difficult task to ensure that one recovers real general relativity as opposed to complex general relativity. Also the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with $\backslash tilde\; =\; \backslash sqrt\; H$. There were serious difficulties in promoting this quantity to a quantum operator. It was Thomas Thiemann who was able to use the generalization of Ashtekar's formalism to real connections ($\backslash beta$ takes real values) and in particular devised a way of simplifying the original Hamiltonian, together with the second term, in 1996. He was also able to promote this Hamiltonian constraint to a well defined quantum operator within the loop representation.〔''Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity'', T. Thiemann'', Phys.Lett. B380 (1996) 257-264.〕 For an account of these developments see John Baez's homepage entry, ''The Hamiltonian Constraint in the Loop Representation of Quantum Gravity''.〔''The Hamiltonian Constraint in the Loop Representation of Quantum Gravity'', http://math.ucr.edu/home/baez/hamiltonian/hamiltonian.html〕 Smolin and others independently discovered that there exists in fact a Lagrangian formulation of the theory by considering the self-dual formulation of the tetradic Palatini action principle of general relativity.〔J. Samuel. ''A Lagrangian basis for Ashtekar's formulation of canonical gravity''. Pramana J. Phys. 28 (1987) L429-32〕〔T. Jacobson and L. Smolin. ''The left-handed spin connection as a variable for canonical gravity.'' Phys. Lett. B196 (1987) 39-42.〕〔T. Jacobson and L. Smolin. ''Covariant action for Ashtekar's form of canonical gravity''. Class. Quant. Grav. 5 (1987) 583.〕 These proofs were given in terms of spinors. A purely tensorial proof of the new variables in terms of triads was given by Goldberg〔''Triad approach to the Hamiltonian of general relativity.'' Phys. Rev. D37 (1987) 2116-20.〕 and in terms of tetrads by Henneaux et al.〔M. Henneaux, J.E. Nelson and C. Schomblond. ''Derivation of Ashtekar variables from tetrad gravity.'' Phys. Rev. D39 (1989) 434-7.〕 ==Further reading== * 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「Ashtekar variables」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.