Summary
The relativistic KMS condition introduced by Bros and Buchholz provides a link between quantum statistical mechanics and quantum field theory. We show that for the P(ø)2 model at positive temperature, the two-point function for fields satisfies the relativistic KMS condition.
Keywords
- Selfadjoint Operator
- Quantum Statistical Mechanic
- Space Translation
- Euclidean Measure
- Reconstruction Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H. Araki: A lattice of von Neumann algebras associated with the quantum theory of a free Bose field. J. Math. Phys. 4:1343–1362 (1963).
H. Araki: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. Res. Int. Math. Soc. (RIMS) 9:165–209 (1973).
H. Araki: Positive cone, Radon-Nikodym theorems, relative Hamiltonian and the Gibbs condition in statistical mechanics. An application of Tomita-Takesaki theory. In C*-algebras and their applications to Statistical Mechanics and Quantum Field Theory (ed.: D. Kastler). North Holland, 1976.
L. Birke and J. Fröhlich: KMS, etc. Rev. Math. Phys. 14:829–871 (2002).
O. Bratteli and D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics I, II. Springer-Verlag, New York, Heidelberg, Berlin, 1981.
J. Bros and D. Buchholz: Towards a relativistic KMS condition. Nucl. Phys. B 429:291–318 (1994).
J. Bros and D. Buchholz: Axiomatic analyticity properties and representations of particles in thermal quantum field theory, New problems in the general theory of fields and particles. Ann. l’Inst. H. Poincaré 64:495–521 (1996).
J. Derezinski, V. Jaksic and C.-A. Pillet: Perturbation theory of W*-dynamics, Liouvilleans and KMS states. Rev. Math. Phys. 15:447–48. (2003).
J. Fröhlich: The reconstruction of quantum fields from Euclidean Green’s functions at arbitrary temperatures. Helv. Phys. Acta 48:355–363 (1975).
J. Fröhlich: Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint. Adv. in Appl. Math. 1:237–256 (1980).
C. Gérard and C.D. Jäkel: Thermal quantum fields with spatially cut-off interactions in 1+1 space-time dimensions. To appear in Journal of Funct. Anal., 2005.
C. Gérard and C.D. Jäkel: Thermal quantum fields without cutoffs in 1+1 spacetime dimensions. arXiv math-ph/0403047. To appear in Rev. Math. Phys.
J. Glimm and A. Jaffe: Quantum Physics, A Functional Point of View. Springer, 1981.
R. Haag, N.M. Hugenholtz and M. Winnink: On the equilibrium states in quantum statistical mechanics. Comm. Math. Phys. 5:215–236 (1967).
R. Haag, D. Kastler and E.B. Trych-Pohlmeyer: Stability and equilibrium states. Comm. Math. Phys. 38:173–193 (1974).
E.P. Heifets and E.P. Osipov: The energy momentum spectrum in the P(ϕ)2 quantum field theory. Comm. Math. Phys. 56:161–172 (1977).
R. Höegh-Krohn: Relativistic quantum statistical mechanics in two-dimensional space-time. Comm. Math. Phys. 38:195–224 (1974).
C.D. Jäkel: Decay of spatial correlations in thermal states. Ann. l’Inst. H. Poincaré 69:425–440 (1998).
C.D. Jäkel: The Reeh-Schlieder property for thermal field theories. J. Math. Phys. 41:1745–1754 (2000).
R.V. Kadison and J.R. Ringrose: Fundamentals of the Theory of Operator Algebras II. Academic Press, New York, 1986.
A. Klein and L. Landau: Stochastic processes associated with KMS states. J. Funct. Anal. 42:368–428 (1981).
A. Klein and L. Landau: Construction of a unique selfadjoint generator for a symmetric local semigroup. J. Funct. Anal. 44:121–137 (1981).
R. Kubo: Statistical mechanical theory of irreversible processes I. J. Math. Soc. Jap. 12:570–586 (1957).
P.C. Martin and J. Schwinger: Theory of many-particle systems. I. Phys. Rev. 115:1342–1373 (1959).
M. Reed and B. Simon: Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-adjointness. Academic Press, 1975.
B. Simon: The P(ϕ)2 Euclidean (Quantum) Field Theory. Princeton University Press, 1974.
R.F. Streater and A.S. Wightman: PCT, Spin and Statistics and all that. Benjamin, New York, 1964.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag
About this chapter
Cite this chapter
Gérard, C., Jäkel, C.D. (2007). On the Relativistic KMS Condition for the P(ø)2 Model. In: de Monvel, A.B., Buchholz, D., Iagolnitzer, D., Moschella, U. (eds) Rigorous Quantum Field Theory. Progress in Mathematics, vol 251. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7434-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7434-1_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-7433-4
Online ISBN: 978-3-7643-7434-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)