# On The Stability Of KMS States In Perturbative Algebraic Quantum Field Theories

Nicolò Drago, F. Faldino, N. Pinamonti

Published 2018 · Physics, Mathematics

We analyze the stability properties shown by KMS states for interacting massive scalar fields propagating over Minkowski spacetime, recently constructed in the framework of perturbative algebraic quantum field theories by Fredenhagen and Lindner (Commun Math Phys 332:895, 2014). In particular, we prove the validity of the return to equilibrium property when the interaction Lagrangian has compact spatial support. Surprisingly, this does not hold anymore, if the adiabatic limit is considered, namely when the interaction Lagrangian is invariant under spatial translations. Consequently, an equilibrium state under the adiabatic limit for a perturbative interacting theory evolved with the free dynamics does not converge anymore to the free equilibrium state. Actually, we show that its ergodic mean converges to a non-equilibrium steady state for the free theory.

This paper references

QFT on curved spacetimes : axiomatic framework and examples

I. M. Ryzhik I. S. Gradshteyn (2007)

Perturbative Algebraic Quantum Field Theory at Finite Temperature

Falk Lindner (2013)

10.1007/s00220-003-0837-1

On the Renormalization Group in Curved Spacetime

S. Hollands (2003)

10.1115/1.3452897

Table of Integrals, Series, and Products

I. S. Gradshteyn (1943)

10.1007/s002200050004

Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

Romeo Brunetti (2000)

10.1007/BF02100096

Micro-local approach to the Hadamard condition in quantum field theory on curved space-time

Marek J. Radzikowski (1996)

10.1016/S0550-3213(02)00059-7

Asymptotic dynamics of thermal quantum fields

J. Bros (2002)

10.1007/BF01026494

Entropy production and its positivity in nonlinear response theory of quantum dynamical systems

I. Ojima (1988)

10.1016/0370-1573(87)90121-9

Real- and imaginary-time field theory at finite temperature and density

N. P. Landsman (1987)

10.1007/s00220-003-0815-7

The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory

Romeo Brunetti (2001)

10.1007/bfb0025525

Perturbation Expansions in Axiomatic Field Theory

O. Steinmann (1971)

10.1142/2364

Thermal field theory

M. Bellac (1996)

10.2977/prims/1195192744

Relative hamiltonian for faithful normal states of a von neumann algebra

H. Araki (1973)

10.1007/978-3-319-09949-1

Perturbative algebraic quantum field theory

Klaus Fredenhagen (2012)

10.1007/978-3-662-09089-3

Operator algebras and quantum statistical mechanics

Ola Bratteli (1979)

10.1007/s00220-014-2141-7

Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics

Klaus Fredenhagen (2014)

10.1063/1.4939955

Quantum field theory on curved spacetimes: Axiomatic framework and examples

K. Fredenhagen (2016)

10.4310/ATMP.2009.V13.N5.A7

Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

Romeo Brunetti (2009)

10.1142/S0129055X04002266

CAUSAL PERTURBATION THEORY IN TERMS OF RETARDED PRODUCTS, AND A PROOF OF THE ACTION WARD IDENTITY

M. Duetsch (2004)

10.1016/0370-2693(90)91749-2

Infrared problem in gφ4 theory at finite temperature

T. Altherr (1990)

10.1007/s00220-002-0719-y

Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime

S. Hollands (2002)

The generally covariant locality principle : A new paradigm for local quantum physics

K. Fredenhagen Brunetti (2003)

10.1007/S002200200602

Non-Equilibrium Steady States of Finite¶Quantum Systems Coupled to Thermal Reservoirs

V. Jaksic (2002)

10.1007/BF01646342

On the equilibrium states in quantum statistical mechanics

R. Haag (1967)

10.1023/A:1018618704438

Natural Nonequilibrium States in Quantum Statistical Mechanics

D. Ruelle (1999)

Local Quantum Physics: Fields, Particles, Algebras

R. Haag (1992)

10.1007/s002200100540

Local Wick Polynomials and Time Ordered Products¶of Quantum Fields in Curved Spacetime

S. Hollands (2001)

10.1007/s00023-016-0521-6

The Generalised Principle of Perturbative Agreement and the Thermal Mass

Nicolò Drago (2015)

10.1007/s00220-008-0670-7

The Time Slice Axiom in Perturbative Quantum Field Theory on Globally Hyperbolic Spacetimes

B. Chilian (2009)

Winnink, “On the equilibrium states in quantum statistical mechanics,

R. Haag (1967)

Rejzner, “Perturbative algebraic quantum field theory,

K. K. Fredenhagen (2012)

10.1007/BF01940765

Stability properties and the KMS condition

Ola Bratteli (1978)

10.1007/978-3-642-02780-2_5

Quantum Field Theory on Curved Backgrounds

Romeo Brunetti (2009)

Trych-Pohlmeyer, “Stability and Equilibrium States,

R. Haag (1974)

The Role of locality in perturbation theory

H. Epstein (1973)

10.1007/BF01651541

Stability and equilibrium states

R. Haag (1974)

10.1007/BF01646264

Return to equilibrium

D. Robinson (1973)

10.1007/BF01044241

Entropy production and nonequilibrium stationarity in quantum dynamical systems. Physical meaning of van Hove limit

I. Ojima (1989)

10.1007/3-540-53862-3_177

Entropy production and nonequilibrium stationarity in quantum dynamical systems

I. Ojima (1991)

Rejzner, “QFT on curved spacetimes: axiomatic framework and examples,

K. K. Fredenhagen (2014)

10.1007/BF02099626

The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes

R. Brunetti (1996)

10.1007/BF02108335

Perturbative quantum field theory at positive temperatures: An axiomatic approach

O. Steinmann (1995)

10.1007/S002200000339

On Entropy Production in Quantum Statistical Mechanics

V. Jaksic (2001)

This paper is referenced by

O ct 2 01 8 Thermal state with quadratic interaction

Nicolò Drago (2018)

10.1007/s00023-019-00859-3

Equilibrium States in Thermal Field Theory and in Algebraic Quantum Field Theory

João Braga de Góes Vasconcellos (2019)

Non-equilibrium steady states for the interacting Klein-Gordon field in 1+3 dimensions

Thomas-Paul Hack (2018)

10.1007/s00023-018-0730-2

Relative Entropy and Entropy Production for Equilibrium States in pAQFT

Nicolò Drago (2017)

10.1007/s00023-018-0739-6

Thermal State with Quadratic Interaction

Nicolò Drago (2017)

10.1007/s00220-020-03775-4

Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence

Sergio Doplicher (2019)

10.1007/s11005-017-0947-x

On the adiabatic limit of Hadamard states

Nicolò Drago (2016)

Thermodynamics of Quantum Fields in Nonstationary Spacetimes

Klaus Fredenhagen (2018)

10.15167/BRAGA-DE-GOES-E-VASCONCELLOS-JOAO_PHD2019-12-10

Thermal equilibrium states in perturbative Algebraic Quantum Field Theory in relation to Thermal Field Theory

J. B. D. G. E. Vasconcellos (2019)