2016/12/11 by Claus Gerhardt, Gerhardt, Claus
Mathematics · Physics and Astronomy · #83 #83C #83C45 #Advanced Differential Geometry Research #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #gr-qc #hep-th #math-ph #math.MP #msc:83 #msc:83C #msc:83C45 #quant-ph
paper · pdf · doi:10.48550/arxiv.1612.03469
18 pages, v2: Added a lemma, Lemma 2.4, and clarified a few points in Section 2
openalex publication_date 2016/12/11 · arxiv created 2017/01/20 · arxiv updated 2017/01/23 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
In our model of quantum gravity the quantum development of a Cauchy hypersurface is governed by a wave equation derived as the result of a canonical quantization process. To find physically interesting solutions of the wave equation we employ the separation of variables by considering a temporal eigenvalue problem which has a complete countable set of eigenfunctions with positive eigenvalues and also a spatial eigenvalue problem which has a complete set of eigendistributions. Assuming that the Cauchy hypersurface is asymtotically Euclidean we prove that the temporal eigenvalues are also spatial eigenvalues and the product of corresponding eigenfunctions and eigendistributions, which will be smooth functions with polynomial growth, are the physically interesting solutions of the wave equation. We consider these solutions to describe the quantum development of the Cauchy hypersurface.