Modular theory and KMS states

Cyclic means M Omega is dense; separating means a Omega equal to zero forces a equal to zero. On that dense set define S-zero by sending a Omega to a-star Omega. Separation makes the prescription well-defined, while the adjoint conjugates scalars, so the operator is anti-linear. It is closable; its closure S has polar decomposition J times the positive square root of Delta. Delta is generally an unbounded positive self-adjoint injective operator, and J is an antiunitary involution.

Tomita–Takesaki's theorem states that J conjugates M onto its commutant and the unitary powers of Delta preserve M by conjugation. Thus sigma-t of a, defined using Delta to the it, is an automorphism of M for every real t. The resulting group is appropriately operator-topology continuous, not generally norm-continuous. This theorem is stated here rather than proved: closability, unbounded domains, and invariance of the algebra are the central analysis, not formal consequences of writing a polar decomposition.

We use the following KMS convention, matching the cited modular-theory source. For a σ-weakly continuous one-parameter group alpha-t of *-automorphisms of M and inverse parameter beta positive, require for every a,b in M a bounded scalar function analytic in the strip with imaginary part between zero and beta, continuous on its closure, whose lower boundary is phi(alpha-t(a)b) and upper boundary is phi(b alpha-t(a)). With this ordering the modular group of a faithful normal state satisfies the condition at beta one. Reversing the placement of the evolving observable changes the associated imaginary-time sign.

For entire analytic elements, the boundary condition can be written phi(alpha at i-beta of a times b)=phi(ba). That shorthand is not valid for an arbitrary bounded a unless its analytic continuation exists in the required domain. The general strip statement avoids pretending every operator has a value at complex time. A suitable invariant analytic *-subalgebra is ultraweakly dense for modular flow. Physical time and its temperature normalization must also be specified before identifying this canonical parameter with a laboratory evolution.

For M_n(C) with an invertible density matrix rho, represent the algebra by left multiplication on Hilbert–Schmidt matrices and take Omega=rho to the one half. This vector is cyclic and separating. The modular operator sends X to rho X rho inverse, and modular automorphisms send a to rho to the it times a times rho to the minus it. All matrix elements are entire analytic, so complex-time calculations here have no hidden domain problem.

If rho is a scalar multiple of the identity, the state is tracial and modular flow is trivial even though M_n can be noncommutative. Conversely, the modular group of a faithful normal state is trivial exactly when the state is tracial. A rank-deficient density matrix cannot be inserted into the inverse-rho formula on the full algebra. One must change the support or representation appropriately; calling its ordinary GNS vector separating would contradict the null positive elements detected in earlier lessons.