Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables

TL;DR

This paper develops an integrable quantum model related to high-energy QCD, constructing the Baxter Q-operator and eigenfunctions using the Quantum Inverse Scattering Method and Separation of Variables.

Contribution

It introduces a novel SL(2,C) spin magnet model, constructs its Baxter Q-operator, and applies Sklyanin's method to find eigenfunctions, advancing the understanding of high-energy QCD states.

Findings

Constructed the R-matrix for SL(2,C) representations.

Explicitly built the Baxter Q-operator for the model.

Derived integral representations for eigenfunctions.

Abstract

We analyze a completely integrable two-dimensional quantum-mechanical model that emerged in the recent studies of the compound gluonic states in multi-color QCD at high energy. The model represents a generalization of the well-known homogenous Heisenberg spin magnet to infinite-dimensional representations of the SL(2,C) group and can be reformulated within the Quantum Inverse Scattering Method. Solving the Yang-Baxter equation, we obtain the R-matrix for the SL(2,C) representations of the principal series and discuss its properties. We explicitly construct the Baxter Q-operator for this model and show how it can be used to determine the energy spectrum. We apply Sklyanin's method of the Separated Variables to obtain an integral representation for the eigenfunctions of the Hamiltonian. We demonstrate that the language of Feynman diagrams supplemented with the method of uniqueness provide a powerful technique for analyzing the properties of the model.