12.1. Vertex operators In order to prove that the ground states | of the free bosonic...

12.1. Vertex operators In order to prove that the ground states | α of the free bosonic theory are obtained by applying the vertex operators Vα(z,¯0) to the conformal vacuum | 0 it is necessary to show that Vα(0,0) is an eigenstate of π0 with eigenvalue α and, furthermore, that anVα(0,0) | 0 = 0, con n > 0. To this aim

a. Prove the validity of the formula [B, eA] = eA [B,A]
assuming that the commutator [B,A] is a constant.

b. Pose B = π0, A = iαϕ(z, ¯ z) and show that [π0,Vα] = αVα
Consequently π0Vα(0,0) | 0 = αVα(0,0) | 0

c. Show that [an,Vα(z, ¯ z)] = α znVα(z, ¯ z)
and conclude that an annihilates Vα(0,0) | 0 .