modular theory (changes) in nLab

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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.

nPOV definition

The modern approach to defining the modular automorphism group is through the theory of noncommutative L_p-spaces?. This was pioneered by Haagerup in 1979 and Yamagami in 1992.

In this approach, given a von Neumann algebra MM, a faithful semifinite normal weight μ\mu on MM, and an imaginary number tt, the modular automorphism associated to MM, μ\mu, and tt is

σ μ t:M→M,m↦μ tmμ −t.\sigma_\mu^t\colon M\to M,\qquad m\mapsto \mu^t m \mu^{-t}.

This approach makes it easy to deduce various properties of the modular automorphism group.

For more details, see a MathOverflow answer.

Traditional definition

Let ℋ\mathcal{H} be a Hilbert space, ℳ\mathcal{M} a von Neumann-algebra with commutant ℳ′\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 A∈ℳA \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 ℳ\mathcal{M}:

   Ad(Δ it)ℳ=ℳfor allt∈ℝ Ad(\Delta^{it}) \mathcal{M} = \mathcal{M} \; \; \text{for all} \; t \in \mathbb{R}

6. JJ maps ℳ\mathcal{M} to ℳ′\mathcal{M}'.

References

• Uffe Haagerup, L pL^p-spaces associated with an arbitrary von Neumann algebra. Algèbres d’opérateurs et leurs applications en physique mathématique. Colloques Internationaux du Centre National de la Recherche Scientifique 274, 175–184.

• Shigeru Yamagami, Algebraic aspects in modular theory, Publications of the Research Institute for Mathematical Sciences 28:6 (1992), 1075-1106. doi.

• Shigeru Yamagami, Modular theory for bimodules, Journal of Functional Analysis 125:2 (1994), 327-357. doi.

General

Introduction:

• Stephen J. Summers: “Tomita-Takesaki Modular Theory” (arXiv)

Role in algebraic quantum field theory:

• Hans-Jürgen Borchers, On Revolutionizing of Quantum Field Theory with Tomita’s Modular Theory, ESI Preprint 773 (1999) [[pdf](https://www.mat.univie.ac.at/~esiprpr/esi773.pdf)]

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

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

• Alain Connes, Carlo Rovelli, Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories, arXiv:gr-qc/9406019, pdf

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)

See also

• Wikipedia, Tomita–Takesaki theory

Modular flow

On Tomita-Takesaki modular flow as emergent time evolution in quantum physics (AQFT):

• Roberto Longo, The emergence of time (arxiv:1910.13926)

Videos of lecture series on modular theory by Masamichi Takesaki and Serban Stratila:

• Master Class in Modular Theory

Noncommutative Integration

A very detailed overview of modular flow, non-commutative L pL_p-spaces, etc. which includes many further references:

• Ryszard Paweł Kostecki W *W^{*}-algebras and noncommutative integration arXiv:1307.4818