  This paper implements the idea of considering the instantonic creation of
brane worlds whose five-dimensional bulk contains a negative cosmological
constant and a scalar quintessence field with time-dependent equation of state,
restricting to the case that the quintessence field couples minimally to
Hilbert-Einstein gravity. We construct an Euclidean formalism, both for the
four- and five-dimensional cases, singling out a Hamiltonian constraint that
depends on the parameter defining the quintessence state equation. Specializing
at several particular values of that parameter, we obtain solutions to the
constraint equation and analyse them both classically and quantum mechanically.
It is found that these solutions can represent either asymptotically anti-de
Sitter wormholes or pure anti-de Sitter spaces whose quantum states are
obtained by means of the Wheeler de Witt equation. Starting with the different
five-dimensional solutions, an instantonic procedure is applied to describe the
creation of geometrically equivalent inflating de Sitter branes whose quantum
states are also evaluated in some cases. We thus consider the quantum state of
the universe to be contributed by all the instantonic paths that correspond to
these particular brane worlds.
