In the definition of vertex algebra, we call the vertex operator state-field correspondence, does that mean that it is an injective map??
Are there some physical interpretations about state-field correspondence ? Or why we need state-field correspondence in physical viewpoint?? 
Does it have some relations to highest weight representations?



About state-field correspondence
Yes, the state-field map $v\mapsto Y(v,z)$ is an injective map, since by the axioms of VOA, $Y(v,z)1|\_{z=0}=v$. 


The state-field correspondence appears in 2-dimensional field theory because such a field theory attaches an amplitude to a "pair of pants" (a 2-sphere with 3 holes). Namely, if you regard two of the holes as "incoming" and one as "outgoing"
(i.e., you regard the surface as a worldsheet of an interaction of two strings in which they unite into one), then to this surface corresponds an operator $A: H\otimes H\to H$, where $H$ is the Hilbert space of the theory (attached to the circle). This can be viewed as a map $A: H\to {\rm End}(H)$, which is the state-field correspondence. In conformal field theory, the field map $Y$ above is obtained in the limit when the holes become very small (so the surface becomes the Riemann sphere without 3 points, say, $0,z,\infty$). For instance, in the minimal models $H=\oplus\_{i=0}^n V\_i\otimes V\_i^\ast$, where $V\_i$ are all the irreducible unitary highest weight representations of the Virasoro algebra of central charge $c=1-6/(m+2)(m+3)$, where $m\ge 1$ is an integer. The restriction of $A$ to the summand $V\_0\otimes V\_0^\ast$ at all three holes turns out to be a tensor product $Y\otimes Y^\vee$, where $Y=Y(z): V\_0\to {\rm End}(V\_0)[[z,z^{-1}]]$, which equips $V\_0$ with a structure of a vertex operator algebra. 


