2006/06/23 by Martin Bojowald, Aureliano Skirzewski · 97 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Black Holes and Theoretical Physics #Classical mechanics #Curvature #Geometry #Gravitation #Hamiltonian (control theory) #Hamiltonian constraint #Loop quantum gravity #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum gravity #Quantum mechanics #Theoretical physics #Wheeler–DeWitt equation #astro-ph #gr-qc #hep-th
paper · pdf · doi:10.1142/s0219887807001941
published in International Journal of Geometric Methods in Modern Physics 04(01), 25-52 (World Scientific) · 28 pages, based on a lecture course at the 42nd Karpacz Winter School of Theoretical Physics ``Current Mathematical Topics in Gravitation and Cosmology,'' Ladek, Poland, February 6-11, 2006
arxiv created 2006/06/23 · openalex publication_date 2007/02/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Effective equations are often useful to extract physical information from quantum theories without having to face all technical and conceptual difficulties. One can then describe aspects of the quantum system by equations of classical type, which correct the classical equations by modified coefficients and higher derivative terms. In gravity, for instance, one expects terms with higher powers of curvature. Such higher derivative formulations are discussed here with an emphasis on the role of degrees of freedom and on differences between Lagrangian and Hamiltonian treatments. A general scheme is then provided which allows one to compute effective equations perturbatively in a Hamiltonian formalism. Here, one can expand effective equations around any quantum state and not just a perturbative vacuum. This is particularly useful in situations of quantum gravity or cosmology where perturbations only around vacuum states would be too restrictive. The discussion also demonstrates the number of free parameters expected in effective equations, used to determine the physical situation being approximated, as well as the role of classical symmetries such as Lorentz transformation properties in effective equations. An appendix collects information on effective correction terms expected from loop quantum gravity and string theory.