Canonical Quantum Gravity
Letting aside temporarily the possibility to unify all forces in a single consistent framework, a question one can ask oneself is whether quantum mechanics can resolve a particular singularity, such as the big bang or that inside a black hole.
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Canonical Quantum Gravity
Tirem-me daqui a metafísica! — Fernando Pessoa, Lisbon Revisited (1923) Spare me metaphysics!
Contents 9.1
Canonical Variables in General Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 First-Order Formalism and Parity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Hamiltonian Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Ashtekar–Barbero Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 ADM Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Wheeler–DeWitt Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Superspace and Quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Semi-classical States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Some Features of Loop Quantum Gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Cosmological Constant Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Chern–Simons State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 as a Condensate?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Problems and Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
408 408 414 424 426 428 429 432 437 439 442 443 447 451 459
Letting aside temporarily the possibility to unify all forces in a single consistent framework, a question one can ask oneself is whether quantum mechanics can resolve a particular singularity, such as the big bang or that inside a black hole. With this goal in mind, we examine an approach to gravity based on the Hamiltonian formalism and its cosmological applications. Section 9.1 is devoted to classical canonical gravity, where the so-called Schwinger, Ashtekar–Barbero and ADM variables are defined. Quantization in ADM variables is discussed in Sect. 9.2, where the Wheeler–DeWitt equation is introduced for the first time. This equation will be the starting point for tackling the big-bang problem in Chap. 10.
© Springer International Publishing Switzerland 2017 G. Calcagni, Classical and Quantum Cosmology, Graduate Texts in Phy
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