A density calculus for quantum field theory

Abstract

These notes provide an introduction to Haagerup noncommutative $L^p$ spaces for quantum field theorists. The aim is to explain how the familiar calculus of density matrices, fractional powers, and Schatten norms extends to local observable algebras of type~III, where an intrinsic trace is unavailable. We develop the construction first for a matrix algebra, making the crossed product, its trace, and the dual action explicit, before passing to general von Neumann algebras. Particular attention is paid to the representation of normal states by Haagerup densities and to the distinction between the crossed-product trace and the canonical Haagerup functional used to compute expectation values. As a concrete application, we calculate the sandwiched Rényi divergence for a coherent excitation of a chiral current, verifying monotonicity under restriction and the Rényi quantum null energy condition at every real order $α>1$. The presentation assumes familiarity with quantum mechanics and elementary quantum field theory, while introducing the operator-algebraic concepts needed for the construction and calculation.