Thermal States and Modular Automorphisms

Ola Bratteli, Derek W. Robinson

Kishore K

We address the problem of time in quantum mechanics: time is an external classical parameter, not a quantum observable, yet quantum gravity requires it to be emergent. We show that a canonical preferred temporal flow arises from within the algebraic structure of quantum mechanics, without presupposing a Hamiltonian, a Schrödinger equation, or a background time parameter. The starting point is a pair (M, ω) — a von Neumann algebra of observables and a faithful normal state — with no other input. From these, Tomita–Takesaki modular theory derives a canonical strongly-continuous one-parameter group of automorphisms σ_t : M → M (the modular flow). The group parameter t labels the spectral flow of the modular operator Δ = S^*S constructed from ω, and is not assumed as background structure. Three results are established. First (Structural Inevitability): any non-tracial quantum system unavoidably carries a non-trivial modular flow — the Tomita–Takesaki modular automorphism group, derived from algebraic data alone. Under the Connes–Rovelli thermal time hypothesis (adopted as an explicit physical postulate, supported by exact agreement with Heisenberg evolution for Gibbs states and with Rindler time via the Bisognano–Wichmann theorem), this flow constitutes physical time. Second (Cocycle Colimit): the directed colimit of all subsystem modular flows exists in the category W*-Dyn, interacting or not; for interacting systems the colimit is twisted by Araki's Connes cocycle, and the assembled flow equals the modular flow of the limiting algebra. Third (Classical Recovery): a structural outline of how, in the decoherence limit under density, separability, and the second law, the framework recovers a separable dense linear order from which n-dimensional spatial structure is derived; full topological reconstruction is left as an open program. This paper is self-contained and requires no companion papers. It extends the Connes–Rovelli thermal time hypothesis and the Page–Wootters relational framework to a universal directed cocycle colimit.

Marcus Berg

Abstract Using recent results in mathematics, I point out that free energies and scale-dependent central charges away from criticality can be represented in compact form where modular invariance is manifest. The main example is the near-critical Ising model on a thermal torus, but the methods are not restricted to modular symmetry, and apply to automorphic symmetries more generally. One application is finite-size effects.

Wolter Groenevelt, Jess Stokman

Roberto Longo, Gerardo Morsella

Heide Narnhofer

For Fermi systems interacting with a Galilei invariant pair potential with a cut-off for particles with highly different velocities the time evolution corresponds to an automorphism. We prove that all states satisfying the KMS-condition are locally normal with respect to the representation in Fock-space. Removing the cut-off limit states remain locally normal provided the interaction is repulsive or of positive type. The modular automorphism of the von Neumann-algebra of the limit state coincides with the time evolution.