The definitions says that third-quantized are fields that create and annihilate second quantized fields. However, there is no a general formalism for third quantized fields established at the level of rigor and applicability as the usual theory of 'second quantized' fields.
A great example that beautifully encodes the virtues and challenges of the formalism is actually string field theory; for an understanding of this analogy in the realm of quantum cosmology, see 'Baby universes, third quantization and the cosmological constant'. The idea is that, even though the 'first quantized' usual strings are on-shell excitations of a second quantized fields, one can tought in the same situation from the worldsheet perspective where the worldsheet CFT is a second quantized theory and a string field can be identified as a 'third quantized' field in the sense that the creation of a usual string actually creates its worldsheet CFT (a second quantized field).
Concerning the potential applications of this formalism to quantum cosmology: One of the main problems of defining a gravitational path integral for a baby universe concerns the field space over one should integrate. The usual work hypothesis says that one can assume that spherically symmetric universes dominate the path integral (at least in the mini-superspace approximation); notice that this hypothesis is automatically satisfied in string field theory as consequence of the perturbative genus-expansion of string theory. On the side of difficulties one can found the fact that it is not obvious how to take non-perturbative effects into account or even the precise identification of a regime in quantum cosmology where 'third quantized fields' can be correctly described by a weakly coupled theory.
