physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
AQFT and operator algebra
homotopical algebraic quantum field theory
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.
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:
$\Delta$ is self-adjoint, positive and invertible (but not bounded).
$\Delta\Omega = \Omega$ and $J\Omega = \Omega$
$J$ is antilinear, $J^* = J, J^2 = \mathbb{1}$, $J$ commutes with $\Delta^{it}$. This implies
For every $A \in \mathcal{M}$ the vector $A\Omega$ is in the domain of $\Delta^{\frac{1}{2}}$ and
The unitary group $\Delta^{it}$ defines a group automorphism of $\mathcal{M}$:
$J$ maps $\mathcal{M}$ to $\mathcal{M}'$.
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