Kiefer C. Quantum gravity

268 QUANTUM COSMOLOGY
from the no-boundary proposal:
df n
f n (0) = 0 , n =2, 3,... , (0) = 0 , n =1.
dτ
This then yields for the wave functions ψ n satisfying the Schrödinger equations
(8.49) a solution that just corresponds to the Euclidean vacuum. The reason is
that essentially the same regularity conditions are required for the no-boundary
proposal and the Euclidean vacuum. In the Lorentzian section, the Euclidean
vacuum for modes with small wavelength (satisfying λ ≪ H −1 )isjustgivenby
the state (8.56); see also Section 10.1. According to the no-boundary proposal,
the multipoles thus enter the Lorentzian regime in their ground state. Because
of its high symmetry, the de Sitter-invariant vacuum was assumed to be a natural
initial quantum state even before the advent of the no-boundary condition
(Starobinsky 1979).
Hawking (1984) has put forward the point of view that the Euclidean path
integral is the true fundamental concept. The fact that a Euclidean metric usually
does not have a Lorentzian section, therefore, does not matter. Only the result—
the wave function—counts. If the wave function turns out to be exponentially
increasing or decreasing, it describes a classically forbidden region. If it is of
oscillatory form, it describes a classically allowed region—this corresponds to the
world we live in. Since one has to use in general complex integration contours
anyway, it is clear that only the result can have interpretational value, with the
formal manipulations playing only the role of a heuristic device.
8.3.3 Tunnelling condition
The no-boundary wave function calculated in the last subsection turned out to be
real. This is a consequence of the Euclidean path integral; even if complex metrics
contribute they should do so in complex-conjugate pairs. The wave function
(8.63) can be written as a sum of semiclassical components of the form exp(iS),
each of which gives rise to a semiclassical world in the sense of Section 5.4
(recovery of the Schrödinger equation). These components become independent
of each other only after decoherence is taken into account; see Section 10.1.
Alternative boundary conditions may directly give a complex wave function,
being of the form exp(iS) in the semiclassical approximation. This is achieved
by the ‘tunnelling proposal’ put forward by Vilenkin; see, for example, Vilenkin
(1988, 2003). 5
The tunnelling proposal is most easily being formulated in minisuperspace.
In analogy with, for example, the process of α-decay in quantum mechanics, it is
proposed that the wave function consists solely of outgoing modes. More generally,
it states that it consists solely of outgoing modes at singular boundaries of
superspace (except the boundaries corresponding to vanishing three-geometry).
In the minisuperspace example above, this is the region of infinite a or φ. What
5 Like the no-boundary proposal, this usually refers to closed three-space Σ. A treatment of
‘tunnelling’ into a universe with open Σ is presented in Zel’dovich and Starobinsky (1984).