The state-field correspondence is a subtle concept, which sometimes puzzles students. In Vertex Operator Algebras, the correspondence is part of the data, and subject to various axioms. If we define states and fields independently, the correspondence is an axiom. If we first define fields, the correspondence could be a definition of states. (A rather trivial definition, since the space of states is the space of fields.)
The state-field correspondence is a subtle concept, which sometimes puzzles students. In Vertex Operator Algebras, the correspondence is part of the data, and subject to various axioms. If we define states and fields independently, the correspondence is an axiom. If we first define fields, the correspondence could be a definition of states. (A rather trivial definition, since the space of states is the space of fields.)
It would be nice to clarify this concept.