nLab
modular theory

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

AQFT and operator algebra

Contents

Idea

This page is about the modular theory introduced by Tomita for von Neumann-algebras. It is important both for the structure theory of von Neumann-algebras and in the Haag-Kastler approach to AQFT, one important example is the Bisognano-Wichmann theorem. It is often called Tomita-Takesaki theory, because the first presentation beyond a preprint is due to Masamichi Takesaki.

Definition

Let \mathcal{H} be a Hilbert space, \mathcal{M} a von Neumann-algebra with commutant \mathcal{M}' and a separating and cyclic vector Ω\Omega. Then there is a modular operator Δ\Delta and a modular conjugation JJ such that:

  1. Δ\Delta is self-adjoint, positive and invertible (but not bounded).

  2. ΔΩ=Ω\Delta\Omega = \Omega and JΩ=Ω J\Omega = \Omega

  3. JJ is antilinear, J *=J,J 2=𝟙J^* = J, J^2 = \mathbb{1}, JJ commutes with Δ it\Delta^{it}. This implies

    Ad(J)Δ=Δ 1 Ad(J) \Delta = \Delta^{-1}
  4. For every AA \in \mathcal{M} the vector AΩA\Omega is in the domain of Δ 12\Delta^{\frac{1}{2}} and

    JΔ 12AΩ=A *Ω=:SAΩ J \Delta^{\frac{1}{2}} A \Omega = A^* \Omega =: SA \Omega
  5. The unitary group Δ it\Delta^{it} defines a group automorphism of \mathcal{M}:

    Ad(Δ it)=for allt Ad(\Delta^{it}) \mathcal{M} = \mathcal{M} \; \; \text{for all} \; t \in \mathbb{R}
  6. JJ maps \mathcal{M} to \mathcal{M}'.

References

Many textbooks on operator algebras contain a chapter about modular theory.

MathOverflow question tomita-takesaki-versus-frobenius-where-is-the-similarity

Discussion in terms of topos theory is in

  • Simon Henry, From toposes to non-commutative geometry through the study of internal Hilbert spaces, 2014 (pdf)

Revised on September 5, 2014 09:59:03 by David Corfield (129.12.18.225)