By P. A. M. DIRAC

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**Example text**

Whilst this is indeed correct for most theories of interest, there exist a few models where this is not the case. These theories are usually called logarithmic theories since, as we shall explain, some of their correlation functions contain logarithms. In this subsection we shall give a brief account of this class of theories; since the general theory has only been developed for theories for which this problem is absent, the present subsection is something of an interlude and not crucial for the rest of this article.

2. Indecomposable Fusion Products and Logarithmic Theories In much of the above discussion we have implicitly assumed that the fusion product of any two irreducible representations of the chiral conformal field theory can be completely decomposed into irreducible representations. Whilst this is indeed correct for most theories of interest, there exist a few models where this is not the case. These theories are usually called logarithmic theories since, as we shall explain, some of their correlation functions contain logarithms.

The above formula is not symmetric under the exchange of φi and φj . In fact, it is manifest from the derivation that (228) must also hold if the comultiplication formulae (226) and (227) are replaced by n n+h−1 m+h−1 ∆u1 ,u2 (Vn (ψ)) = m=1−h n +ε1 l=1−h for n ≥ 1 − h, and ∆u1 ,u2 (V−n (ψ)) = ∞ m=n m−h n−h +ε1 ∞ l=1−h un−m (Vm (ψ) ⊗ 1l) 1 n+h−1 l+h−1 un−l (1l ⊗ Vl (ψ)) , 2 (229) (−u1)m−n (V−m (ψ) ⊗ 1l) n+l−1 n−h −(n+l) (−1)l+h−1 u2 (1l ⊗ Vl (ψ)) , (230) for n ≥ h. e. the quotient of the direct product by the relations that guarantee that ∆ = ∆; this construction is based on an idea of Richard Borcherds, unpublished (see [125]).

### Classical Theory of Radiating Electrons by P. A. M. DIRAC

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