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arXiv:hep-th/9803245

The Relation Between KMS-states for Different Temperatures

Christian D. Jäkel

Published Mar 30, 1998• 0 citations
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Abstract

Given a thermal field theory for some temperature $β^{-1}$, we construct the theory at an arbitrary temperature $ 1 / β'$. Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region $Ø_\circ \subset \rr^4$, but coincide with the given KMS state in the space-like complement of a slightly larger region $\hatØ$. By a weak*-compactness argument there always exists a convergent subnet of states as the size of $ Ø_\circ$ and $ \hatØ$ tends towards $ \rr^4$. Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between the boundaries of $ Ø_\circ$ and $ \hatØ$. We show that this surface energy can be controlled by a generalized cluster condition.

DOI
10.48550/arxiv.hep-th/9803245

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Phase IV closed the free/geometric-flow sector on the third verdict: the registered modular orientation is geometric and irreducible to any CPTP channel, yet it coincides with the Borchers–Wiesbrock spectral condition (energy positivity) — already-known QFT serving as the arrow. That verdict rests on a single structural fact: a KMS state has one modular temperature (Tomita–Takesaki). Any genuinely new arrow beyond the spectral condition therefore requires a state that is not KMS nor generalized-Gibbs. This paper attacks the one corner that is at once outside the third verdict, in scope (no gravity), and tractable: the current-carrying non-equilibrium steady state (NESS). Proceeding symmetrically, with preregistered decision rules frozen before any run: Island A (a Generalized Gibbs Ensemble after an integrable quench) closes null / horn b — the charge tower is absorbed by a single recalibration c(R) → 1, and a finite-size positive is diagnosed as a charge-basis conditioning artefact. The genuine NESS is decided in two stages. On the lattice, the preregistered instrument returns HORN A, grade (ii): the bulk-interval modular Hamiltonian is irreducible to the local-equilibrium-plus-current generator, with a clean law r∞ ≈ Δβ/2. A second, independently frozen preregistration takes the field limit a/β → 0 and returns global collapse: the decisive residual against the Bernard–Doyon boost-of-thermal reference vanishes as ~a² to the equilibrium floor across every drive, while the discriminator's teeth stay intact. The lattice HORN A is reclassified as a UV band-curvature (inflection-point) artefact; the growing distance-to-local-cone that produced it measures the intrinsic non-locality of the Cardy–Tonni boost-of-thermal entanglement Hamiltonian, a known-QFT feature, not new structure. The third verdict extends to the continuum non-KMS, positive-entropy-production sector. Any genuine new modular arrow is pushed to strongly-coupled / holographic NESS, named and left out of scope.