Nonperturbative solutions for canonical quantum gravity: An overview |
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| Institution: | 1. Key Laboratory of Applied Surface and Colloid Chemistry, Ministry of Education, School of Chemistry and Chemical Engineering, Shaanxi Normal University, Xi’An, Shaanxi, 710119, China;2. Key Laboratory of Coal Science and Technology, Ministry of Education and Shanxi Province, Taiyuan University of Technology, Taiyuan 030024, China;1. Department of Biostatistics, Yale School of Public Health, New Haven, CT, USA;2. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, USA;3. Section of Pulmonary, Critical Care and Sleep Medicine, Yale School of Medicine, New Haven, CT, USA;4. Department of Chronic Disease Epidemiology, Yale School of Public Health, New Haven, CT, USA;1. Key Laboratory of Integrated Regulation and Resource Development on Shallow Lakes, Ministry of Education, College of Environment, Hohai University, Nanjing, 210098, China;2. Institute of Blue and Green Development, Shandong University, Weihai, 264209, China;3. SDU Life Cycle Engineering, Department of Chemical Engineering, Biotechnology, and Environmental Technology, University of Southern Denmark, 5230, Odense, Denmark;4. Water@leeds, School of Civil Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom;1. Department of Mechanical Engineering, SCSVMV University, Kancheepuram, Tamil Nadu 631561, India;2. Department of Mechanical Engineering, Sri Sai Ram Engineering College, Chennai, Tamil nadu 600044, India |
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| Abstract: | In this paper we will make a survey of solutions to the Wheeler-De Witt equation which have been found up to now in Ashtekar's formulation for canonical quantum gravity. Roughly speaking they are classified into two categories, namely, Wilson-loop solutions and topological solutions. While the program of finding solutions which are composed of Wilson loops is still in its infancy, it is expected to be developed in the near future. Topological solutions are the only solutions at present which can be interpreted in terms of spacetime geometry. While the analysis made here is formal in the sense that we do not deal with rigorously regularized constraint equations, these topological solutions are expected to exist even in the fully regularized theory and they are considered to yield vacuum states of quantum gravity. We also make an attempt to review the spin network states as intuitively as possible. In particular, the explicit formulae for two kinds of measures on the space of spin network states are given. |
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