行政院國家科學委員會補助專題研究計畫 □X 成果報告
□期中進度報告
（計畫名稱）
非微擾正則重力理論的精確量子態 2/2
Exact Quantum States in Non-perturbative Canonical Gravity 2/2
計畫類別：□X 個別型計畫 □ 整合型計畫
計畫編號：NSC 94－2112－M－006－006
執行期間： 2005 年 8 月 1 日 至 2007 年 1 月 31 日
計畫主持人： 許祖斌 國立成功大學物理系
共同主持人：
計畫參與人員：
本成果報告包括以下應繳交之附件：
□赴國外出差或研習心得報告一份
□X 赴大陸地區出差或研習心得報告一份
□出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
執行單位： 國立成功大學物理系
中 華 民 國 96 年 1 月 25 日
I、中文摘要
本計劃將注重在非微擾正則重力理論的精確 量子態及其物理性質。經由關鍵的推導﹐
我們可將量子重力的super-Hamiltonian 限制條件表達為兩個特定運算元的對異子
(commutator)。對於自旋網路態的 Hilbert 空間﹐這兩個運算元恰好是陳-Simons 和 “自旋網 路的體積” 運算元。此外﹐我們也透過精確量子重力態和 Bekenstein-Hawking 商的公式的 比較算出Immirzi 參數。
關鍵詞: 正則重力理論的精確量子態, super-Hamiltonian 限制條件, 陳-Simons 和 “自旋網路 的體積” 運算元, Immirzi 參數。
Abstract
The research studies the constriants equations and exact solutions of 4-dimensional canonical quantum gravity and their physical properties. A crucial observation permits us to express the super-Hamiltonian constraint as the commutator of two specific operators. For the Hilbert space of spin foam states, the operators are: the Chern-Simons functional, and the regularized 3-hands spin network operator which functions as a well-defined “volume operator”. In addition we estimated the Immirzi parameter for loop quantum gravity by comparing the
Bekenstein-Hawking formula with the entropy from exact pure loop quantum gravity states based upon the SO(3) gauge group.
Keywords :
Non-perturbative quantum gravity states, quantum contraint equations, Chern-Simons operator, Immirzi parameter.
II、緣由與目的
本計畫揭開量子規範場與重力之間所隱含之相互作用﹕
1)注重在非微擾正則重力理論的精確量子態及其物理性質。將量子重力的 super-Hamiltonian 限制條件表達為兩個特定運算元的對異子。
2)透過精確量子重力態和 Bekenstein-Hawking 商的公式的比較算出 Immirzi 參數。
III、結果與討論
I summarize and briefly discussed the major results below:
Major Results :
It is noted that the classical super-Hamiltonian of four-dimensional gravity as simplified by Ashtekar through the use of gauge potential and densitized triad variables can furthermore be succinctly expressed as a vanishing Poisson bracket involving three-geometry elements. This is discussed in the general setting of the Barbero extension of the theory with arbitrary
non-vanishing value of the Immirzi parameter, and when a cosmological constant is also present.
A proposed quantum constraint of density weight two which is polynomial in the basic conjugate variables is also demonstrated to correspond to a precise simple ordering of the operators, and can thus help to resolve the factor ordering ambiguity in the extrapolation from classical to quantum gravity. Alternative expression of a density weight one quantum constraint which may be more useful in the spin network context is also discussed, but this constraint is non-polynomial and is not motivated by factor ordering.
It has long been known that 3-dimensional diffeomorphism invariance would require the quantum states to be functionals of 3-geometries. It may therefore surmised that,
albeit a nontrivial endeavor, it ought to be possible to express the Wheeler-DeWitt Equation of the full theory in terms of explicit 3-geometry elements. However the constraint is also required to be satisfied at each point on the Cauchy surface. Both these requirements are remarkably realized in the reformulation here.
We also argued that the effective gauge group for pure four-dimensional loop quantum gravity (LQG) is SO(3) (or SO(3,C)) instead of SU(2) (or SL(2,C)). As a result, links with half-integer spins in spin network states are not realized for pure LQG, implying a modification of the spectra of area and volume operators. Our observations imply a new value of γ ≈ 0.170 for the Immirzi parameter which is obtained from matching the Bekenstein–Hawking entropy to the number of states from LQG calculations. Moreover, even if the dominant contribution to the entropy is not assumed to come from configurations with the minimum spins, the results of both pure LQG and the supersymmetric extension of LQG can be
made compatible when only integer spins are realized for the former, while the latter also contains half-integer spins, together with an Immirzi parameter for the supersymmetric case which is twice the value of the SO(3) theory.
IV、計畫成果自評
The Chern-Simons functional has served as a fertile link between quantum field theories of three and four dimensions. In General Relativity with Ashtekar variables, it has the additional
significance of being the carrier of information of both intrinsic and extrinsic curvatures The research vindicates that while the volume operator has become a preeminient object in the current manifestation of LQG, the volume element and the Chern-Simons functional can be of equal significance, and need not be mutually exclusive. Both these fundamental objects appear explicitly in the reformulation of the Wheeler-DeWitt constraint. This is an important step in elucidating and vindicating the expected importance and explicit role of both the Chern-Simons and volume operator.
The Immirzi parameter is an ambiguity in the canonical quantization of four-dimensional gravity based upon the Barbero extension. We pointed out that SO(3) should be the correct gauge group for pure loop quantum gravity without matter, andobtained an improved value by matching the Bekenstein-Hawking entropy for large black holes to the non-perturbative states of quantum gravity. We also verify that the −1/2 coefficient of logarithmic correction to the
Bekenstein –Hawking entropy formula is robust, independent of whether only integer, or also half-integer spins, are realized.
Publications/Preprints/Presentations:
[1] C. H. Chou, Y. Ling, C. Soo and H. L. Yu,
Effective gauge group of pure loop quantum gravity is SO(3): New estimate of the Immirzi parameter, Phys. Lett. B 637 (2006) 12-15.
[2]C. Soo, Progress in non-perturbative canonical quantum gravity , invited article, in The Physics Update, Vol. 10 No. 1, (July, 2006), p.17-21, Institute of Physics Singapore.
[3]C. Soo, Simplification of the constrainst of 4-d gravity, in Proceedings of the VIIth. Int. Conf.
on Gravitaiton and Astrophysics (ICGA7), edited by James M Nester, Chiang-Mei Chen &
Jong-Ping Hsu.
[4]C. Soo, Three-Geometry and reformulation of the Wheeler-DeWitt equation, (submitted to Class. Quantum Grav.)
[5]C. Soo, Modern Canonical Gravity I & II, invited series of lectures, Dept. of Physics National U. of Singapore (Jan. 19 & Jan. 25, 2006)
[6]C. Soo, Reformulation of the Wheeler-DeWitt equation, invited presentation, 5th OCPA Meeting, Taipei, (June 29, 2006).
[7]C. Soo, Wheeler-DeWitt Equation and its reformulation, invited seminar, Beijing Normal U.
(July 18, 2006).
[8]C. Soo, Chiral Gravity and the Standard Model, invited seminar, Inst. for Theoretical Physics Chinese Academy of Sciences, Beijing (July 21, 2006)
赴大陸地區出差或研習心得報告
報告人姓名 許祖斌
服務機構 及 現職
國 立 成 功 大 學 物理 系所 教授 時間 95 年 7 月 4 日- 95 年 7 月 30 日
會議
地點 訪問﹕
中國科學院 理論物理所 (北京)
計畫編號 NSC 94-2112-M-006-006
（中文）受邀請訪問 中國科學院理論物理所 會議名稱
（英文）Research visit : Institute of Theoretical Physics, Chinese Academy of Sciences (Beijing).
（中文）學術演講/報告﹕
題目﹕
1)Chiral 重力理論與粒子物理標準模型 (95 年 7 月 21 日﹐中國科學院理論物理所) 2)Wheeler-DeWitt 方程和它的改寫
(95 年 7 月 19 日﹐ 北京師範大學) 發表論文題目
（英文）seminar/lectures:
1) Chiral Gravity and the Standard Model
(July 21, 2006﹐ Inst. Theo. Phys., Chinese Academy of Sciences) 2) Wheeler-DeWitt Equation and its reformulations
(July 19, 2006﹐ Beijing Normal University)
I、參加會議經過、考察參觀活動﹕
受副所長吳岳良教授邀請訪問中國科學院理論物理所﹐做學術研討。
訪問時間是95 年 7 月 5 日到 7 月 30 日。
我分別在7 月 21 日(中國科學院理論物理所) 和 7 月 19 日(北京師範大學) 給學術演講/報告。演講題目分別是“Chiral 重力理論與粒子物理標準模型”
和“Wheeler-DeWitt 方程和它的改寫” 。
此外﹐也參加了95 年 7 月 14-16 日的在中國科學院 Morningside Center of Mathematics 所舉 辦的 Third International ASTROD Symposium on Laser Astrodynamics, Space Test of
Relativity and Gravitational-wave Astronomy 的活動和研討會。
II、與會心得
這一次訪問吳岳良﹐郭漢英﹐菜榮跟﹐黃朝商研究員探討以及學習了 Chiral fermion 理論的詳細 regularization, 彎曲時空 gauged 規範 Poincare 群
的理論以及重力torsion 場的分析。對我的研究非常有幫助﹐也建立了未來合作方向。
在北京師範大學於馬永革教授和當時在場訪問的世界頂尖loop quantum gravity
專家Professor Carlo Rovelli 舉行了一場研討座談會﹐學到了嚴格定義量子重力理論的 super-Hamiltonian Constraint; 對我所研究的 Wheeler-DeWitt 方程的改寫十分有用。
這一次也和馬永革教授建立了Wheeler-DeWitt 方程的改寫和嚴格定義的合作。
此外﹐ASTROD Symposium 也是中-德雙方激光天文動力學研討會﹐演講專家包括許多 位來自于中國和德國的天文物理學和重力波天文物理學家。參加這個活動讓我有機會學到 了這些領域目前及未來最頂尖的研究成果和發展方向。
