Algebraic formulation of local temperature states
Project/Area Number  09640262 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
OJIMA Izumi Kyoto Univ., RIMS,Assoc.Prof., 数理解析研究所, 助教授 (60150322)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,600,000 (Direct Cost : ¥1,600,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  Classification of symmetry breaking patterns / Spontaneous symmetry collapse / Algebraic quantum field theory / Quantum statistical mechanics / Local temperatrue states / Relativistic thermodynamics / 対称性の破れとその分類 / spontaneous symmetry collapse / 代数的場の量子論 / 量子統計力学 / 局所温度平衡状態 / 相対論的熱力学 / 対称性の破れ / Algebraic QFT / Local Quantum Physics / Thermoclynamic Phace / Spontaneous Symmetry Collapse / Tewperature State 
Research Abstract 
To clarify the physical and mathematical meanings of spontaneous symmetry collapse discovered by D.Buchholz and myself in [2], I elaborated in 1997 on fundamental issues concerning relations among thermodynamic phases, order parameters to distinguish different phases, center of represented algebras and symmetry of systems. What I found is useful results involving the spectrum of centre, central ergodicity and approximate sequences of local operators tendeing to central elements, which exhibit interesting roles of the basic notions in quantum probability theory in connection with large deviations when applied to the discussions with low energy theorems. While some of these were reported in [3], I have been elaboraing on basic points in quantumtheoretical extensions of large deviation principle. In the former half of 1998, 1 was invited by Academy of Sciences in Gottingen as Gauss Professor at Gottingen University where I collaborated with Profs.D.Buchholz and H.Roos on the mathematical
… More
formulation of local temperature states in algebraic QFT.I also elaborated upon the classification of symmetry breaking patterns [5], which answers the abovementioned question by placing the result of [2] in a wider perspective with all possible patterns of symmetry breaking exhausted and classified. Concerning the former, its outline was reported at RIMSmeeting of "Mathematics of Quantum Information and Quantum Chaos" and a joint paper is in preparation. The idea is to map a given state to the set of all KMS states via suitable family of local observables in small neighbourhoods of each spacetime point to represent it as a mixture of various KMS states, through which a localized version of thermal equilibrium is attained in a novel way. There is a nontrivial example of such a local temperature state in the massless scalar field model which shows interesting aspects of nonequilibrium suggesting some promising direction of generalrelativistic extension of thermodynamics and quantum statistical mechanicss. In the autumn of 1998, I had discussions with Prof.S.Maumary at Univ.de Lausanne on a homologicalalgebraic reformulation of BRS cohomology and we have been continuing a joint project to complete this work. Less

Report
(4results)
Research Output
(15results)