# Quantum field theory of point particles and strings by Brian Hatfield

By Brian Hatfield

The aim of this booklet is to introduce string concept with out assuming any heritage in quantum box conception. half I of this e-book follows the improvement of quantum box idea for aspect debris, whereas half II introduces strings. Allof the instruments and ideas which are had to quantize strings are built first for element debris. therefore, half I offers the most framework of quantum box idea and gives for a coherent improvement of the generalization andapplication of quantum box concept for element debris to strings.

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Next we consider the transformation of this amplitude under a rotation by 2π; this is implemented by the M¨obius transformation exp(2πiL0), Ω (∞) e2πiL0 µ(z)µ(0) = e− 2πi 4 Ω (∞) µ(e2πi z)µ(0) 1 = z 4 A + B log(z) + 2πiB , (247) where we have used that the transformation property of vertex operators (39) also holds for non-meromorphic fields. On the other hand, because of (246) we can rewrite Ω (∞) e2πiL0 µ(z)µ(0) 1 = z 4 Ω (∞) e2πiL0 ω(0) + log(z)Ω(0) . e. L0 Ω = 0, L0 ω = Ω. Thus we find that the scaling operator L0 is not diagonalisable, but that it acts as a Jordan block 0 1 0 0 (251) on the space spanned by Ω and ω.

However, this does not seem to be correct since the triplet algebra [150] has only finitely many irreducible representations, but contains indecomposable representations in their fusion products that lead to logarithmic correlation functions [151]. Logarithmic conformal field theories are not actually pathological; as was shown in [41] a consistent local conformal field theory that satisfies all conditions of a local theory (including modular invariance of the partition function) can be associated to this triplet algebra.

Conformal Field Theory 30 1 1 (22 + 5c) (m − n) (2m2 − mn + 2n2 − 8) Lm+n , (156) 48 30 where Λk are the modes of a quasiprimary field of conformal weight hΛ = 4. This field is a normal ordered product of L with itself, and its modes are explicitly given as + ∞ Λn = Ln−k Lk + k=−1 −2 k=−∞ Lk Ln−k − 3 (n + 2) (n + 3) Ln . 10 (157) One can check that this set of commutators satisfies the Jacobi-identity. ) Subsequently, various classes of W -algebras have been constructed [79–86]. There have also been attempts to construct systematically classes of W -algebras [87, 88] following [89].