Zassenhaus decomposition of half-sided translations and generalizations in 2d conformal field theory

A bstract We study the half-sided translations associated to Rindler wedge algebras for conformal field theories in 1+1 Minkowski spacetime, generated by an unbounded operator G G, in terms of bilinear forms G, G ′ made from entanglement Hamiltonians of the underlying algebras such that G G = G + G ′. We show that despite entanglement Hamiltonians being ill-defined operators on Hilbert space, G, G ′ can be regularized using smooth bump functions to operators G G ̂, G^ G ̂ ′ with well-defined commutators, and use them to do a centered Zassenhaus expansion of exp (iGs i G s) in terms of G G ̂ and G^ G ̂ ′ which is tractable and respects causality. We show that in fact half-sided translations is a special case in a large class of operators O O for which a similar decomposition can be done by defining O O = O L + O R with O L, O R chosen approriately.