  We give introductions into the representation theory of the Virasoro algebra,
Wightman axioms and vertex algebras in the first part.
  In the second part, we compare the above definitions. We give a proof of
Luescher and Mack that a dilation invariant 2D QFT with an energy-momentum
tensor gives rise to two commuting unitary representations of the Virasoro
algebra.
  In the last chapter we compare Wightman QFTs to vertex algebras. We present
Kac's proof that every Wightman Moebius CFT (a 2D Wightman QFT containing
quasiprimary fields) gives rise to two commuting strongly-generated
positive-energy Moebius conformal vertex algebras. If the number of generating
fields of each conformal weight is finite, then these vertex algebras are
unitary quasi-vertex operator algebras. As a corollary using Luescher--Mack's
Theorem we obtain that a Wightman CFT (a Wightman Moebius CFT with an
energy-momentum tensor) gives rise to a conformal vertex algebra which
furthermore becomes a unitary vertex operator algebra, if the number of
generating fields of each conformal weight is finite. We reverse Kac's
arguments and get a converse proof that two unitary (quasi)-vertex operator
algebras can be combined to give a Wightman (Moebius) CFT.
