Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Quantum cosmology with self–interacting scalar field 197
prime are derivatives with respect to The Hamiltonian can be constructed
and we expressed it as Since N is a Lagrange multiplier,
we have the constraint and follow the Dirac quantization
procedure this equation is known as WDW equation. This
equation is independent of N , however in the following we use the
gauges N = 1 and since we obtain simpler equations with
those choices.
2.1. GAUGE N=1
With the choice N = 1, the action (5) becomes
hence the canonical conjugate momenta corresponding to and and
the corresponding Hamiltonian can be calculated
now, the canonical momenta in Eq. (7) are converted into operators in
the standard way, and the ambiguity
of factor ordering is encoded in the parameter The resulting WDW
equation is
Next we present three simple solvable cases for different cosmological
terms.
Case The solution to WDW equation without a
cosmological term and is
where is a separation constant, the Bessel function order is given by
By superposition of the above solutions we obtain