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

Abstract

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 $\mathcal{G}$, in terms of bilinear forms $G, G'$ made from entanglement Hamiltonians of the underlying algebras such that $\mathcal{G} = G+G'$. We show that despite entanglement Hamiltonians being ill-defined operators on Hilbert space, $G, G'$ can be regularized using smooth bump functions to operators $\hat{G}, \hat{G}'$ with well-defined commutators, and use them to do a centered Zassenhaus expansion of $\exp(i \mathcal{G} s)$ in terms of $\hat{G}$ and $\hat{G}'$ which is tractable and respects causality. We show that in fact half-sided translations is a special case in a large class of operators $\mathcal{O}$ for which a similar decomposition can be done by defining $\mathcal{O} = O_L+O_R$ with $O_{L}, O_{R}$ chosen approriately.