  Many rings and algebras arising in quantum mechanics can be interpreted as
skew PBW (Poincar\'e-Birkhoff-Witt) extensions. Indeed, Weyl algebras,
enveloping algebras of finite-dimensional Lie algebras (and its quantization),
Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum
matrices, among many others, are examples of skew PBW extensions. In this paper
we extend the classical Ore and Goldie theorems, known for skew polynomial
rings, to this wide class of non-commutative rings. As application, we prove
the quantum version of the Gelfand-Kirillov conjecture for the skew quantum
polynomials.
