Feynman Path Integral Formulation

5.1 Cosmological Wavefunctions 143
Physically the above expression means that the total quantum-mechanical amplitude
for a particle to arrive at (x,t) is a sum over all possible values x ′ of the total amplitude
to arrive at the (x ′ ,t ′ ) (which is given by Ψ(x ′ ,t ′ ), multiplied by the amplitude
to go from x to x ′ , which is given by the propagator G(x,t;x ′ ,t ′ ). The propagator
itself corresponds to a special situation: the amplitude where the particle started out
at precisely (x ′ ,t ′ ).
Let us recall that in Feynman’s formulation of quantum mechanics the propagator
G is expressed as a sum over all paths connecting initial and final points, weighted
by an action I,
∫ x
{ }
f (t f )
ī
G(x f ,t f ;x i ,t i ) ≡〈x f ,t f |x i ,t i 〉 = [dx(t)] exp
h I[x(t)] . (5.4)
x i (t i )
The paths x(t) contributing to the integral are known to be continuous, but not necessarily
differentiable (one can give arguments in support of the statement that differentiable
paths have measure zero), which requires in general that the above integral
be carefully defined on a lattice of N points with spacing a, with the limit a → 0,
N → ∞ taken at the end.
Returning to the gravitational case, the question arises then of how to compute
the path integral in Eq. (5.2), even in the absence of matter, and what boundary
conditions need to be imposed. In gravity the analogue of Eq. (5.4) is the quantum
mechanical amplitude
∫ ( j) g
〈g ( f )
ij
,φ ( f ) |g (i)
ij ,φ(i) 〉 = [dg
g (i)
ij ,φ (i)
ij ,φ ( f )
{
μν ][dφ] exp − 1¯h }
Î(g μν,φ)
, (5.5)
where the functional integral is over all four-geometries that match the initial (i)
and final ( f ) field configurations on the two spacelike surfaces. One noteworthy
aspect of such gravitational amplitudes is the fact that, since all intervening fourgeometries
are summed over, there is no notion of unique intervening proper time
interval: the proper time distance between the two hypersurfaces will depend on the
specific choice of interpolating four-geometry in the ensemble.
As mentioned previously, in computing the ground state wave functional Ψ of
Eq. (5.2) the proposal has been put forward to functionally integrate over all metrics
associated with compact Euclidean four-geometries specified by g μν , with a given
three-metric g ij on the boundary. For obvious reasons this is usually referred to as
the “no-boundary” proposal. It elegantly bypasses the issue of having to specify a
boundary or continuity condition on cosmological singularities, by suitably restricting
the choice of geometries at “initial” times. In this approach the wave functional
for pure gravity is given by (from now on we set again ¯h = 1)
∫
Ψ[g ij ]= [dg μν ] exp { −Î(g μν ) } , (5.6)
M
with an Euclidean action containing both volume (M) and boundary (∂M) terms,