Modular theory for the von Neumann algebras of Local Quantum Physics

TL;DR

This paper explores the modular theory of von Neumann algebras in Local Quantum Physics, covering second quantization, the Bisognano-Wichmann theorem, and reconstruction theorems for symmetry groups and free fields.

Contribution

It introduces a modular framework for von Neumann algebras in quantum field theory, linking algebraic structures with physical symmetries and field constructions.

Findings

Demonstrates the Bisognano-Wichmann theorem for Wightman fields

Shows how modular groups can reconstruct symmetry actions

Constructs free field algebras from symmetry representations

Abstract

In the first part, the second quantization procedure and the free Bosonic scalar field will be introduced, and the axioms for quantum fields and nets of observable algebras will be discussed. The second part is mainly devoted to an illustration of the Bisognano-Wichmann theorem for Wightman fields and in the algebraic setting, with a discussion on the physical meaning of this result. In the third part some reconstruction theorems based on modular groups will be described, in particular the possibility of constructing an action of the symmetry group of a given theory via modular groups, and the construction of free field algebras via representations of the symmetry group.