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 Ĝ G ̂ , Ĝ G ̂ ′ with well-defined commutators, and use them to do a centered Zassenhaus expansion of exp ( iGs i G s ) in terms of Ĝ G ̂ and Ĝ 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.