Modular Operators and Entanglement in Supersymmetric Quantum Mechanics

TL;DR

This work links the Tomita–Takesaki modular operator framework to supersymmetric quantum mechanics by constructing von Neumann algebras from SUSY operators and identifying the modular conjugation $J$ as a physically meaningful, anti-linear quantity that quantifies bipartite entanglement via concurrence. The approach yields explicit results: concurrence in bipartite SUSY supermultiplets equals $C(|\\Phi_k\\rangle)=|\\langle\\Phi_k|J|\\Phi_k\\rangle|$, and modular maps $J$ between algebras encode duality between SUSY sectors. Applications to graphene's 2D Dirac fermions and to the Jaynes–Cummings model demonstrate concrete mappings between SUSY Hamiltonians and their modular images, including anti-JC forms with non-RWA terms. By bridging operator-algebra methods with entanglement measures, the paper provides a unifying framework with potential extensions to topological defects and algebraic quantum field theory contexts.

Abstract

The modular operator approach of Tomita-Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system. An explicit operator characterizing the dual infinite degeneracy structure of a supersymmetric two dimensional system is given by the modular conjugation operator. Furthermore, the entanglement of formation for these supersymmetric systems using concurrence is shown to be related to the expectation value of the modular conjugation operator in an entangled bi-partite supermultiplet state thus providing a direct physical meaning to this anti-unitary, anti-linear operator as a quantitative measure of entanglement. Finally, the theory is applied to the case of two-dimensional Dirac fermions, as is found in graphene, and a supersymmetric Jaynes Cummings Model.