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

TL;DR

The paper addresses half-sided translations in 1+1 dimensional CFTs by decomposing the generator $\mathcal{G}$ into $G+G'$, where $G$ and $G'$ are built from entanglement Hamiltonians and are initially ill-defined as operators. It introduces smooth bump-function regularizations to obtain $\hat{G},\hat{G}'$ with well-defined nested commutators and derives a centered Zassenhaus expansion for $e^{i\mathcal{G}s}$, yielding a tractable decomposition in terms of $\hat{G},\hat{G}'$ and their commutators while preserving causality. The work extends the construction to a larger class of operators $\mathcal{O}=O_L+O_R$ under suitable splitting, thereby providing a formal framework for stitching subregion modular flows in 2d CFTs and informing potential applications to AdS/CFT, JT gravity, and related holographic settings. These results illuminate how regularization can render otherwise ill-defined entanglement constructs operational for dynamical, region-specific transformations in quantum 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.

Figures (4)

• Figure 2: Rindler wedges and regions 1,2,3; the condition \ref{['halfs']} is satisfied by algebras of the regions 1 and 2.
• Figure 3: The OPE of two stress tensor insertions $T(x) T(y)$ converges for $x, y > \beta_{\mathcal{U}}$ where $\beta_{\mathcal{U}}$ is chosen such that all excitations to produce $\chi$ lie before $x<\beta_\mathcal{U}$.
• Figure 4: First derivative of $R(x)$ as a smoothed function with a 5th-degree polynomial on $(-1,1)$ with $\epsilon = 0.2$ shown in red; the original $R(x)\propto x$ having a constant derivative is shown in black.
• Figure 5: Second derivative of $R(x)$ as a smoothed function with a 5th-degree polynomial on $(-1,1)$ with $\epsilon = 0.2$.