The No Boundary Proposal (Hartle and Hawking)
The path integral for quantum gravity should be taken over all compact
Euclidean metrics.
One can paraphrase this as The Boundary Condition Of The Universe Is That It Has No
Boundary.
In the rest of this lecture I shall show that this no boundary proposal seems to account
for the universe we live in. That is an isotropic and homogeneous expanding universe with
small perturbations. We can observe the spectrum and statistics of these perturbations in
the fluctuations in the microwave background. The results so far agree with the predictions
of the no boundary proposal. It will be a real test of the proposal and the whole Euclidean
quantum gravity program when the observations of the microwave background are extended
to smaller angular scales.
In order to use the no boundary proposal to make predictions, it is useful to introduce
a concept that can describe the state of the universe at one time.
Consider the probability that the spacetime manifold M contains an embedded three
dimensional manifold Σ with induced metric hij. This is given by a path integral over all metrics gab on M that induce hij on Σ. If M is simply connected, which I will assume, the surface Σ will divide M into two parts M+ and M −.
In this case, the probability for Σ to have the metric hij can be factorized. It is the product of two wave functions Ψ+ and Ψ −. These are given by path integrals over all metrics on M + and M − respectively, that induce the given three metric hij on Σ. In most cases, the two wave functions will be equal and I will drop the superscripts + and −. Ψ is called 43
M +
Σ
_
M
Z
Probability of induced metric hij on Σ =
d[ g] e−I
metrics on M that
induce hij on Σ
M +
Σ
Probability of hij = Ψ+( hij) × Ψ −( hij )
Z
where Ψ+( hij) =
d[ g] e−I
metrics on M + that
induce hij on Σ
the wave function of the universe. If there are matter fields φ, the wave function will also depend on their values φ 0 on Σ. But it will not depend explicitly on time because there is no preferred time coordinate in a closed universe. The no boundary proposal implies
that the wave function of the universe is given by a path integral over fields on a compact
manifold M+ whose only boundary is the surface Σ. The path integral is taken over all metrics and matter fields on M+ that agree with the metric hij and matter fields φ 0 on Σ.
One can describe the position of the surface Σ by a function τ of three coordinates xi on Σ. But the wave function defined by the path integral can’t depend on τ or on the choice 44
of the coordinates xi. This implies that the wave function Ψ has to obey four functional differential equations. Three of these equations are called the momentum constraints.
Momentum Constraint Equation
∂Ψ
= 0
∂hij ; j
They express the fact that the wave function should be the same for different 3 metrics
hij that can be obtained from each other by transformations of the coordinates xi. The fourth equation is called the Wheeler-DeWitt equation.
Wheeler – DeWitt Equation
∂ 2
1
G
3
ijkl
− h 2 R Ψ = 0
∂hij∂hkl
It corresponds to the independence of the wave function on τ . One can think of it as the Schrödinger equation for the universe. But there is no time derivative term because the
wave function does not depend on time explicitly.
In order to estimate the wave function of the universe, one can use the saddle point
approximation to the path integral as in the case of black holes. One finds a Euclidean
metric g 0 on the manifold M+ that satisfies the field equations and induces the metric hij on the boundary Σ. One can then expand the action in a power series around the
background metric g 0.
1
I[ g] = I[ g 0] +
δgI 2 δg + …