nLab modular theory




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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: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}'.




  • 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]

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 March 30, 2023 at 07:36:30. See the history of this page for a list of all contributions to it.