這篇論文探討重力和純量場在球對稱的簡化，並詳細地推導和討論簡化後的變數、限制條件和它們的特性。對於量子化的重力場和純量場的Ashtekar-Wheeler-DeWitt 方程式被證實可以簡化為描述量子化純量場隨Schwinger-Tomonaga 時間演進的薛丁格方程。如果特定的條件成立，在de Sitter度規暴脹的宇宙，Schwinger-Tomonaga 時間隨著宇宙的體積增加。假設典型的位能小於浦朗克能量密度、實際上的量子態偏離Chern-Simons 態的程度小，以及純量場的位能近似於常數，這個從Ashtekar-Wheeler-DeWitt 方程式推導的宇宙暴脹理論出現的相當自然。

Spherically symmetric reduction of gravitational and scalar fields is investigated. The reduced variables, constraints and their properties are derived and discussed at length. It is shown that the Ashtekar-Wheeler-DeWitt Equation for quantised gravitational and scalar fields reduces to a Schrodinger Equation which describes quantised scalar field evolving with respect to a Schwinger-Tomonaga time which increases with the volume of the universe in an inflationary universe with approximate de Sitter metric if certain conditions are satisfied. This inflationary scenario emerges rather naturally from the Ashtekar-Wheeler-DeWitt Equation provided the typical potential is small compared to the Planck energy density, the departure of the actual state from the Chern-Simons state is small, and the potential of the scalar field is approximately constant.

Bibliography
[1] Abroe M. E. et al., Correlations Between the WMAP and MAXIMA Cosmic Microwave Background Anisotropy Maps, Astrophys. J. 605 607 (2004)
[2] Guth A. H., The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D23 347a (1981)
[3] Linde A. D., A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys.Lett.B 108 389 (1982)
[4] Albrecht A. and Steinhardt P. J., Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys.Rev.Lett.48 1220 (1982)
[5] Starobinsky A., Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys.Lett.B 117 175 (1982)
[6] S. Perlmutter et al., Cosmology from Type Ia supernovae, Bull.Am.Astron.Soc. 29 1351 (1997)
[7] S. Perlmutter et al.,Discovery of a supernova explosion at half the age of the Universe and its cosmological implications, Nature 391 51 (1998)
[8] Ashtekar A., New Hamiltonian Formulation of General Relativity, Phys. Rev. D36 1587 (1987)
[9] Ashtekar A., New Variables for Classical and Quantum Gravity, Phys. Rev. Lett. 57 2244 (1986)
[10] Kodama H., Holomorphic wavefunction of the universe, Phys. Rev. D42 2548 (1990)
[11] Soo C., Wavefunction of the universe and Chern-Simons perturbation theory, Class. Quant. Grav. 19 1051 (2002)
[12] Soo C., Ashtekar-Wheeler-De Witt equation and inflationary scenario, in Proceedings of the 10th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic field Theories (MG X MMIII) 20-26 Jul (2003), ed. Mario Novello, Santiago Perez Bergliaffa, World Scientific 2005
[13] DeWitt B. S., Quantum Theory of Gravity. 1. The Canonical Theory, Phys. Rev. 160 1113 (1967)
[14] Wheeler J. A., Superspace and the quamtum geometrodynamics, in Battelle rencontres Lectures in mathematics and physics, ed. DeWitt C. M. and Wheeler J. A., W.A.Benjamin, New York (1968)
[15] Arnowitt R., Deser S. and Misner C., Canonical variables for general relativity, Phys. Rev. 117 1595 (1960)
[16] Dirac P. A. M., Lectures on quantum mechanics, Courier Dover Publications, Originally published: Belfer Graduate School of Science, Yeshiva University , New York (1964)
[17] Chou C. Y., Closure of constraint algebra and its application to spherically symmetric quantum gravity, MSc. Thesis National Chung Kung University (June. 2008)
[18] Wei P. A., A study of the inner product and reality conditions of self-dual quantum gravity, MSc. Thesis National Chung Kung University (June. 2008)
[19] See, for instance, Soo C., Introduction to Classical and Quantum Canonical Gravity, and Classical and Quantum Gravity with Ashtekat Variables: Case Study of Chern-Simons State, Lectures I-III, Center for Advanced Studies, National Tsinghua University, Beijing (Nov.2007)
[20] Forgacs P. and Manton N. S., Space-Time Symmetries in Gauge Theories, Commun. Math. Phys. 72 15 (1980)
[21] Jacobson T. and Smolin L., Covariant Action for Ashtekar's Form of Canonical Gravity, Class. Quant. Grav. 5 583 (1988)
[22] Alvarez E., Quantum gravity: an introduction to some recent results, Rev. Mod. Phys. 61 561 (1989)
[23] Tomonaga S., On a relativistically invariant formulation of the quantum theory of wave fields, Prog. Theor. Phys. 1 27 (1946)
[24] Schwinger J. S., Quantum electrodynamics. I: A covariant formulation, Phys. Rev. 74 1439(1948)