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Symmetries in classical and quantum treatment of Einstein’s cosmological equations and mini-superspace actions. (English) Zbl 1392.83004

Summary: The use of automorphisms of the various Bianchi-type Lie algebras as Lie-point symmetries of the corresponding Einstein field equations entails a reduction of their order and ultimately leads to the entire solution space. When a valid reduced action principle exists, the symmetries of the configuration mini-supermetric space can also be used, in conjunction with the constraints, to provide local or non-local constants of motion. At the classical level, depending on their number, these integrals can even secure the acquisition of the entire solution space without any further solving of the dynamical equations. At the quantum level, their operator analogues can be used, along with the Wheeler-DeWitt equation, to define unique wave functions that exhibit singularity-free behavior at a semi-classical level.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
34C14 Symmetries, invariants of ordinary differential equations
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
17B81 Applications of Lie (super)algebras to physics, etc.
83C45 Quantization of the gravitational field
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