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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 $M$, a faithful semifinite normal weight $\mu$ on $M$, and an imaginary number $t$, the modular automorphism associated to $M$, $\mu$, and $t$ is

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

For more details, see a MathOverflow answer.

Traditional definition

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

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

2. $\Delta\Omega = \Omega$ and $J\Omega = \Omega$

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

   $$Ad(J) \Delta = \Delta^{-1}$$

4. For every $A \in \mathcal{M}$ the vector $A\Omega$ is in the domain of $\Delta^{\frac{1}{2}}$ and

   $$J \Delta^{\frac{1}{2}} A \Omega = A^* \Omega =: SA \Omega$$

5. The unitary group $\Delta^{it}$ defines a group automorphism of $\mathcal{M}$:

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

6. $J$ maps $\mathcal{M}$ to $\mathcal{M}'$.

References

• Uffe Haagerup, $L^p$-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 $L_p$-spaces, etc. which includes many further references:

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