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  "body": "\n## Summary\n\nTomita–Takesaki theory starts from a [[von Neumann algebra|PSFWQVZ0FS]] with a cyclic and separating vector.\nIt attaches to it a positive self-adjoint modular operator $\\Delta$ and an antiunitary modular conjugation $J$.\nIts main theorem says that $J$ turns the algebra into its [[commutant|1VC7EJ1ZBZ]].\nIt also says that the unitaries $\\Delta^{it}$ give a one-parameter group of automorphisms, the modular automorphism group.\nSo every faithful normal state comes with a natural time evolution.\nThe theory is the basis of Connes' classification of type $\\mathrm{III}$ factors and of the use of KMS states in quantum statistical mechanics.\n\n## Explanations\n\n### L1 human\n\nFor a trace, the order of a product does not matter: $\\tau(xy) = \\tau(yx)$.\nMost states are not traces, and Tomita–Takesaki theory measures exactly how far a state is from being one.\nThe answer is a flow $\\sigma_t$ of automorphisms of the algebra that repairs the order of a product.\nFor a matrix algebra with the state $\\varphi(x) = \\operatorname{tr}(\\rho x)$ given by an invertible density matrix $\\rho$, this flow is $\\sigma_t(x) = \\rho^{it} x \\rho^{-it}$.\nThen $\\varphi(xy) = \\varphi(y \\, \\rho x \\rho^{-1})$, so moving $x$ to the other side costs the twist $x \\mapsto \\rho x \\rho^{-1}$.\nFor a trace, $\\rho$ is a multiple of the identity and the flow does nothing.\n",
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