Quantum Gravity

QUANTUM THEORY OF COLLAPSING DUST SHELLS 227t =(u + v)/2, r =(−u + v)/2, (7.87)p t = p u + p v , p r = −p u + p v . (7.88)The constraint function then assumes the form p u p v =(p 2 t − p 2 r)/4. Upon quantization,one obtains the operator −ˆp t which is self-adjoint and has a positivespectrum, −ˆp t ϕ(p) =pϕ(p), p ≥ 0. It is the generator of time evolution andcorresponds to the energy operator E ≡ M. Sincer is not a Dirac observable, itcannot directly be transformed into a quantum observable. It turns out that thefollowing construction is useful,ˆr 2 := − √ p d2 1 √pdp 2 . (7.89)This is essentially a Laplacian and corresponds to a concrete choice of factorordering. It is a symmetric operator which can be extended to a self-adjointoperator. In this process, one is naturally led to the following eigenfunctions ofˆr 2 :√2pψ(r, p) := sin rp , r ≥ 0 . (7.90)πOne can also construct an operator ˆη that classically would correspond to thedirection of motion of the shell.The formalism has now reached a stage in which one can start to studyconcrete physical applications. Of particular interest is the representation of theshell by a narrow wave packet. One takes at t = 0 the following family of wavepackets:ψ κλ (p) ≡ (2λ)κ+1/2√ p κ+1/2 e −λp , (7.91)(2κ)!where κ is a positive integer, and λ is a positive number with dimension of length.By an appropriate choice of these constants, one can prescribe the expectationvalue of the energy and its variation. A sufficiently narrow wave packet can thusbe constructed.One can show that the wave packets are normalized and that they obeyψ κλ (p) =ψ κ1 (λp) (‘scale invariance’). The expectation value of the energy iscalculated as∫ ∞dp〈E〉 κλ ≡0 p pψ2 κλ (p) , (7.92)with the result〈E〉 κλ = κ +1/2 . (7.93)λIn a similar way, one finds for the variation√ 2κ +1∆E κλ = . (7.94)2λSince the time evolution of the packet is generated by −ˆp t , one has