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 $G,G^{\prime }$ made from entanglement Hamiltonians of the underlying algebras such that $\mathcal{G}=G+G^{\prime }$. We show that despite entanglement Hamiltonians being ill-defined operators on Hilbert space, $G,G^{\prime }$ can be regularized using smooth bump functions to operators $\hat{G},{\hat{G}}^{\prime }$ 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 ${\hat{G}}^{\prime }$ 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.

major revision with new results