Quantum Gravity

150 QUANTUM GEOMETRODYNAMICSIn analogy to the ground-state wave function of the quantum-mechanical harmonicoscillator, the ground-state solution for (5.42) reads(Ψ 0 [φ] =det 1/4 ω) (exp − 1 ∫π 2)d 3 xd 3 x ′ φ(x)ω(x, x ′ )φ(x ′ ), (5.43)with the ground-state energy given byE 0 = 1 2 trω = 1 ∫d 3 xd 3 x ′ ω(x, x ′ )δ(x − x ′ )= 1 ∫V22 (2π) 3d 3 pω(p) . (5.44)This is just the sum of the ground-state energies for infinitely many harmonicoscillators. Not surprisingly, it contains divergences: the infrared (IR) divergenceconnected with the spatial volume V (due to translational invariance) and theultraviolet (UV) divergence connected with the sum over all oscillators. This isthe usual field-theoretic divergence of the ground-state energy and can be dealtwith by standard methods (e.g. normal ordering). Note that the normalizationfactor in (5.43) is also divergent:(det 1/4 ω) ( 1=expπ 4 tr ln ω ) ( ∫ √ )V=expπ 32π 3 d 3 p2 + mp ln2. (5.45)πOne can define the many-particle states (the Fock space) in the usual mannerthrough the application of creation operators on (5.43). The divergence (5.45)cancels in matrix elements between states in Fock space. However, the space ofwave functionals is much bigger than Fock space. In fact, because there is nounique ground state in the case of time-dependent external fields, any Gaussianfunctional is called a ‘vacuum state’, independent of whether it is the groundstate of some Hamiltonian or not. A general Gaussian is of the formΨ Ω [φ] =det 1/4 (ΩRπ)exp(− 1 ∫2)d 3 xd 3 x ′ φ(x)Ω(x, x ′ )φ(x ′ ) , (5.46)where Ω ≡ Ω R +iΩ I is in general complex and time-dependent. One can definein the usual manner an annihilation operator (skipping the integration variablesfor simplicity)A = √ 1 ∫ (Ω −1/2RΩφ + δ )(5.47)2 δφand a creation operatorA † = √ 1 ∫ (Ω −1/2RΩ ∗ φ − δ ). (5.48)2 δφOne has the usual commutation relation [A(x),A † (y)] = δ(x − y), and the vacuumstate (5.46) is annihilated by A, AΨ Ω =0.