Feynman Path Integral Formulation

4.9 Connection with the Feynman Path Integral 117
as well as Eqs. (4.77) and (4.78). It was DeWitt who first showed that these are
equivalent to the Schrödinger equation for quantized matter field in a classical gravitational
background g ij (x),
i ∂ ∂t Φ(t) =Ĥφ [g ij ]Φ(t) , (4.83)
with a background g ij -dependent matter field Hamiltonian
∫
}
Ĥ φ [g ij ]= d 3 x
{N(x)Ĥ φ (x)+N i (x)Ĥ φ i (x) . (4.84)
A discussion on how the Wheeler-deWitt equation and its semiclassical expansion
relate to the conventional covariant Feynman diagram picture can be found
in (Barvinsky and Kiefer, 1998; Barvinsky, 1998).
4.9 Connection with the Feynman Path Integral
In principle any solution of the Wheeler-deWitt equation corresponds to a possible
quantum state of the universe. It is also clear the effects of the boundary conditions
on the wavefunction will act to severely restrict the class of possible solutions. In
ordinary quantum mechanics these are determined by the physical context of the
problem and some set of external conditions. In the case of the universe as a whole
the situation is less clear, and in many approaches some suitable set of boundary
conditions are postulated instead, based on general arguments involving concepts
such as simplicity or economy.
One proposal (Hartle and Hawking, 1983) is to restrict the quantum state of the
universe by requiring that the wave function Ψ be determined by a path integral over
compact Euclidean metrics. The wave function would then be given by
∫
Ψ[g ij ,φ] = [dg μν ][dφ] exp ( −Î[g μν ,φ] ) , (4.85)
where Î is the Euclidean action for gravity plus matter
Î = − 1 ∫
16πG
d 4 x √ g(R − 2λ) − 1 ∫
8πG
d 3 x √ ∫
g ij K −
d 4 x √ gL m . (4.86)
The functional integral would be restricted to those four-metrics which have the induced
metric g ij and the matter field φ as given on the boundary surface S. One
would expect, as is already the case in non-relativistic quantum mechanics where
the path integral with a boundary surface satisfies the Schrödinger equation (Feynman
and Hibbs, 1963), that the wavefunction constructed in this way would also
automatically satisfy the Wheeler-DeWitt equation. In (Hartle and Hawking, 1983)
this is shown to be indeed the case.