Kiefer C. Quantum gravity

BOUNDARY CONDITIONS 269
does ‘outgoing’ mean? The answer is clear in quantum mechanics, since there
one has a reference phase ∝ exp(−iωt). An outgoing plane wave would then have
a wave function ∝ exp(ikx). But since there is no external time t in quantum
cosmology, one can call a wave function ‘outgoing’ only by definition (Zeh 1988).
In fact, the whole concept of tunnelling loses its meaning if an external time is
lacking (Conradi 1998).
We have seen in (5.22) that the Wheeler–DeWitt equation possesses a conserved
‘Klein–Gordon current’, which here reads
j = i 2 (ψ∗ ∇ψ − ψ∇ψ ∗ ) , ∇j = 0 (8.65)
(∇ denotes again the derivatives in minisuperspace). A WKB solution of the
form ψ ≈ C exp(iS) leadsto
j ≈−|C| 2 ∇S. (8.66)
The tunnelling proposal states that this current should point outwards at large
a and φ (provided, of course, that ψ is of WKB form there). If ψ were real (as
is the case in the no-boundary proposal), the current would vanish.
In the above minisuperspace model, we have seen that the eikonal S(a, φ),
which is a solution of the Hamilton–Jacobi equation, is given by the expression
(cf. (8.63))
S(a, φ) = (a2 V (φ) − 1) 3/2
. (8.67)
3V (φ)
Wewouldthushavetotakethesolution∝ exp(−iS) since then j would, according
to (8.66), become positive and point outwards for large a and φ. For
a 2 V > 1, the tunnelling wave function then reads
(
ψ T ∝ (a 2 V (φ) − 1) −1/4 exp − 1 ) (
exp −
i
)
3V (φ) 3V (φ) (a2 V (φ) − 1) 3/2 ,
(8.68)
while for a 2 V < 1 (the classically forbidden region), one has
(
ψ T ∝ (1 − a 2 V (φ)) −1/4 exp − 1 (
1 − (1 − a 2 V (φ)) 3/2)) . (8.69)
3V (φ)
As for the inhomogeneous modes, the tunnelling proposal also picks out the
Euclidean vacuum.
8.3.4 Comparison of no-boundary and tunnelling wave function
An important difference between the no-boundary and the tunnelling condition
is the following: whereas the tunnelling condition is imposed in the oscillatory
regime of the wave function, the no-boundary condition is implemented in the
Euclidean regime; the oscillatory part of the wave function is then found by a
matching procedure. This leads in the above example to the crucial difference