[Paper Review] Temparature as Order Parameter of Broken Scale Invariance

Izumi Ojima|ArXiv.org|Nov 16, 2003

Quantum Mechanics and Applications8 references4 citations

TL;DR

This paper establishes inverse temperature β as a macroscopic order parameter for classifying mutually disjoint thermal sectors in algebraic quantum field theory, arising from broken scale invariance under renormalization-group transformations. By introducing an augmented C*-algebra with a non-trivial center, the paper shows that β parametrizes distinct KMS states through continuous center characters, unifying thermal, quantum, and geometric aspects via categorical adjunctions and modular theory.

ABSTRACT

In algebraic quantum field theory the (inverse) temperature is shown to be a macroscopic extit{order parameter} to parametrize mutually disjoint thermal extit{sectors} arising from the extit{broken scale invariance} under renormalization-group transformations. This is accomplished in a mathematical formalism for the consistent treatment of extit{explicitly broken symmetries} such as broken scale invariance, on the basis of a clear-cut criterion for the symmetry breakdown in a unified scheme for sectors proposed recently by the author.

Motivation & Objective

• To establish a unified mathematical framework for describing thermal, quantum, and geometric aspects in quantum field theory.
• To clarify how broken scale invariance under renormalization-group transformations gives rise to distinct thermal sectors.
• To identify inverse temperature β as a macroscopic order parameter distinguishing mutually disjoint KMS states.
• To extend the DHR-DR superselection theory to include spontaneously broken symmetries and non-equilibrium states.
• To provide a categorical, algebraic formulation of order parameters using center characters of augmented C*-algebras.

Proposed method

• Introduce an augmented C*-algebra  to accommodate the center characterizing broken scale invariance and thermal order parameters.
• Use the GNS representation πβ associated with a KMS state ωβ to define von Neumann algebras M := πβ(A)′′.
• Apply interpolation theory in non-commutative Lp-spaces to define virtual temperatures τ = β/p, with 0 ≤ τ ≤ β.
• Employ categorical adjunctions to relate generic states ω to reference states ωρ via selection criteria and parameterized comparison.
• Utilize the relative modular operator Δφ1,φ0 and faithful normal semifinite weight φ0 to define α-divergences in Lp and Lq spaces.
• Analyze the KMS condition under scale transformations to verify that β shifts continuously, confirming its role as an order parameter.

Experimental results

Research questions

• RQ1 How can inverse temperature β be rigorously identified as an order parameter in thermal quantum field theory?
• RQ2 In what way does broken scale invariance under renormalization-group flow lead to mutually disjoint thermal sectors?
• RQ3 How can the center of a C*-algebra encode macroscopic order parameters for thermal states?
• RQ4 What is the role of virtual temperatures τ = β/p in interpolating between quantum statistical and geometric structures?
• RQ5 How does the proposed framework unify superselection sectors across vacuum, thermal, and non-equilibrium quantum field theories?

Key findings

• The inverse temperature β is shown to be a macroscopic order parameter that parametrizes mutually disjoint thermal sectors arising from broken scale invariance.
• The center of the universal representation Â, particularly its spectrum, distinguishes sectors via continuous center characters, contrasting with the discontinuous disjointness of representations at the W*-level.
• The KMS condition under scale transformations leads to a parameter shift in β, confirming its role as an order parameter in the renormalization group flow.
• Virtual temperatures τ = β/p (0 ≤ τ ≤ β) emerge naturally in non-commutative Lp-spaces, reflecting kinematic limitations in accessing lower temperatures without dynamical considerations.
• The framework allows extension of information-geometric concepts such as α-divergence, relative entropy, and Fisher information to infinite-dimensional quantum systems via Lp-pairings.
• The augmented algebra  enables a unified treatment of thermal, quantum, and geometric aspects through categorical adjunctions and center-based classification of states.