More on doubled Hilbert space in double-scaled SYK

TL;DR

This paper explores the doubled Hilbert space formalism in the double-scaled SYK model, establishing a relationship between the 1-particle Hilbert space and the doubled 0-particle Hilbert space through a linear isomorphism, and clarifies the roles of entangler and disentangler.

Contribution

It reveals the connection between the 1-particle Hilbert space and the doubled 0-particle Hilbert space via a linear isomorphism and identifies the entangler and disentangler as duals of this isomorphism.

Findings

The 1-particle Hilbert space is related to the doubled 0-particle Hilbert space by a linear isomorphism.

The entangler and disentangler are given by the duals of the inverse and the isomorphism itself.

The study advances understanding of the doubled Hilbert space structure in SYK models.

Abstract

We continue our study of the doubled Hilbert space formalism of double-scaled SYK model initiated in [arXiv:2401.07403]. We show that the 1-particle Hilbert space introduced by Lin and Stanford in [arXiv:2307.15725] is related to the doubled Hilbert space $\mathcal{H}_0\otimes \mathcal{H}_0$ of the 0-particle Hilbert space $\mathcal{H}_0$ by some linear isomorphism $\mathcal{C}$. It turns out that the entangler and disentangler appeared in our previous study are given by the dual (or adjoint) of $\mathcal{C}^{-1}$ and $\mathcal{C}$.