Feynman Path Integral Formulation

126 4 Hamiltonian and Wheeler-DeWitt Equation
In concluding the discussion on minisuperspace models as tools for studying
the physical content of the Wheeler-DeWitt equation it seems legitimate to ask the
following question: to what extent can results for such models, and specifically the
ones discussed in the previous section, be representative of what might or might not
happen in the full quantum theory
To such purpose let us consider a field theory model that is non-trivial, yet much
simpler that gravity: a self-coupled quantum scalar field φ(x) in four dimensions
with Lagrangian
L φ = 1 2 (∂ μ φ) 2 − 1 2 m2 φ 2 − 1 4 λφ4 , (4.120)
with a mass term proportional to m 2 and a quartic self-interaction proportional to
the dimensionless coupling λ. The perturbative treatment in four dimensions (Wilson,
1973; for a review see Zinn-Justin, 2002) shows that the quantum theory is
strongly interacting at short distances, whereas the long distance behavior is determined
by the Gaussian fixed point at λ = 0. The theory essentially becomes
non-interacting at large distances, with an effective running coupling λ(p 2 ) ∼ p 2 →0
1/|log(p 2 /Λ 2 )|→0.
A minisuperspace approximation to such a theory would consist in neglecting
altogether all spatial derivatives of the quantum field, by setting ∇φ = 0. The model
is then described by a single degree of freedom φ(t), with Lagrangian
L φ = 1 2 ˙φ 2 − 1 2 m2 φ 2 − 1 4 λφ4 . (4.121)
From the expression for the momentum variable π φ = ˙φ, and the substitution ˆπ φ →
(¯h/i)∂/∂φ, one then obtains the quantum Schrödinger equation for stationary states
in this reduced phase space, which is
{
− 1 ∂ 2
}
2 ∂φ 2 + U(φ) − k Ψ(φ) =0 , (4.122)
with U(φ)= 1 2 m2 φ 2 + 1 4 λφ4 and k the energy eigenvalue. Normalizable solutions
are of course the usual Hermite eigenfunctions of the quantum harmonic oscillator
in the position representation. The ground state wavefunction (with k = 1 2m) would
then roughly correspond to the minisuperspace solution of the Wheeler-DeWitt
equation HΨ = 0 for gravity.
The shortcomings of such a minisuperspace truncation of the original quantum
theory of Eq. (4.120) are now becoming evident: the model no longer contains any
propagating degrees of freedom along the spatial directions x. Since φ is assumed
to be constant in x, any correlations in the spatial directions are absent, which is
troubling since for the free part one knows that such vacuum correlations are not
negligible: they are given by
〈φ(x,0)φ(x ′ ,0)〉 ∼
1
|x − x ′ | 2 , (4.123)