Doubled Hilbert space in double-scaled SYK

TL;DR

This paper develops a doubled Hilbert space framework for matter correlators in the double-scaled SYK model, showing that the intersection-counting structure is captured by a similarity transformation generated by an entangler E and its inverse. It rewrites correlators as overlaps ⟨0,0|X⟩ with X mapped to |X⟩ ∈ H⊗H, and provides explicit constructions for the two-point and crossed four-point functions, highlighting how q-deformed oscillator dynamics and the thermo-field double state organize the computation. The key technical result is that the intersection-counting operator is conjugated by E and E^{-1} (or E_{Δ1} and its inverse in the crossed case), enabling a compact, tensor-network–inspired description of DSSYK matter correlators. The approach clarifies the bulk-grounding interpretation of chord diagrams, connects to MERA-like entanglement structures, and opens paths toward relating this framework to other two-sided formalisms such as Lin:2023trc, with future work focusing on symmetry realizations and broader applications of the doubled-space method.

Abstract

We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space $\mathcal{H}\otimes\mathcal{H}$, where $\mathcal{H}$ is the Fock space of $q$-deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain ``entangler'' and ``disentangler''. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.