modular theory




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

AQFT and operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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 Haagerup and Yamagami in 1992 Yamagami.

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:MM,mμ 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 \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}'.


  • 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.

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


An introduction into Tomita-Takesaki modular theory is here:

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

…while a paper that puts it to serious work is this:

  • H.J. Borchers: “On Revolutionizing of Quantum Field Theory with Tomita’s Modular Theory”, (ESI preprint page)

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)

See also

Modular flow

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

Last revised on July 28, 2021 at 16:37:59. See the history of this page for a list of all contributions to it.