>>11786461A quantum field is an operator-valued map , where is (Minkowski) spacetime, is the Fock space corresponding to the type of field (e.g. properly symmetrized or antisymmetrized if bosonic or fermionic), and is the space of linear operators on .
The term "wavefunction" is, IMO, defunct terminology, but typically it refers to a Hilbert space element which can be viewed as a smooth map , where is a real, smooth manifold. Since all separable Hilbert spaces of the same dimension are isomorphic, we must have that . Thus there are many ways to represent a quantum state, and the term "wavefunction" typically refers to the representation (made formal by defining the proper algebra of observables which define the map domain, i.e. the "physical space" ).
That said, many people extend "wavefunction" to the finite-dimensional case as well, in which case they are invoking Riesz duality, treating the Hilbert space element as a linear map on its dual space, .
