Systems of classical particles in the grand canonical ensemble, scaling limits and quantum field theory

Reviews in Mathematical Physics

Volumn 17, Issue 2, 2005, Pages 175-226

  Albeverio, Sergio ^a   Gottschalk, Hanno ^a   Yoshida, Minoru W ^b  

^a UNIVERSITY OF BONN   (Germany)

^b UNIVERSITY OF TOKYO   (Japan)

Author keywords

Cluster expansion; Correlation functionals; Euclidean quantum field theory; Extremal Gibbs measures; Local interactions; Particle systems in the grand canonical ensemble; Poisson random fields; Potts and Widom Rowlinson models

Indexed keywords

EID: 17144408445     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129055X05002327     Document Type: Article

References (62)

1. S. Albeverio, J. E. Fenstad, R. Høegh-Krohn and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Pure and Applied Math., 122 (Academic Press, New York, 1987).
2. S. Albeverio and H. Gottschalk, Scattering theory for quantum fields with indefinite metric, Commun. Math. Phys. 216 (2001) 491.
3. S. Albeverio, H. Gottschalk and J.-L. Wu, Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions, Rev. Math Phys. 8(6) (1996) 763.
4. S. Albeverio, H. Gottschalk and J.-L. Wu, Models of local relativistic quantum fields with indefinite metric (in all dimensions), Commun. Math. Phys. 184 (1997) 509.
5. S. Albeverio, H. Gottschalk and J.-L. Wu, Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric, Phys. Lett. B405 (1997) 243.
6. S. Albeverio, H. Gottschalk and J.-L. Wu, Scattering behavior of quantum vector fields obtained from Euclidean covariant SPDEs, Rep. Math. Phys. 44(1/2) (1999) 21.
7. S. Albeverio, H. Gottschalk and M. W. Yoshida, Representing Euclidean Quantum fields as scaling limits of particle systems, J. Stat. Phys. 108(1/2) (2002) 631-639.
8. S. Albeverio, Z. Haba and F. Russo, A two-dimensional, semi-linear heat equation perturbed by white noise, Probab. Theory Relat. Fields 121 (2001) 319-366.
9. S. Albeverio and R. Høegh-Krohn, Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non-polynomial interactions. Commun. Math. Phys. 30 (1973) 171.
10. S. Albeverio and R. Høegh-Krohn, The scattering matrix for some non-polynomial interactions I, Helv. Physica Acta 46 (1973) 504.
11. S. Albeverio and R. Høegh-Krohn, Euclidean Markov fields and relativistic quantum fields from stochastic partial differential equations, Phys. Lett. B177 (1986) 175.
12. S. Albeverio and R. Høegh-Krohn, Quaternionic non-Abelian relativistic quantum fields in four space-time dimensions, Phys. Lett. B189 (1987) 329.
13. S. Albeverio and R. Høegh-Krohn, Construction of interacting local relativistic quantum fields in four space-time dimensions, Phys. Lett. B200 (1988) 108-114, with erratum in ibid. B202 (1988) 621.
14. S. Albeverio, R. Høegh-Krohn, H. Holden and T. Kolsrud, Representation and construction of multiplicative noise, J. Funct. Anal. 87 (1989) 250.
15. S. Albeverio, K. Iwata and T. Kolsrud, Random fields as solutions of the inhomogenous quarternionic Cauchy-Riemann equation. I. Invariance and analytic continuation, Commun. Math. Phys. 132 (1990) 550.
16. S. Albeverio and J.-L. Wu, Euclidean random fields obtained by convolution from generalized white noise, J. Math. Phys. 36 (1995) 5217-5245.
17. G. Battle, Wavelets and Renormalization (World Scientific, Singapore/ New Jersy/London/Hong Kong, 1999).
18. H. Bauer, Mass- und Integrationstheorie (W. de Gruyter, Berlin/New York, 1990).
19. C. Becker, R. Gielerak and P. Lugievicz, Covariant SPDEs and quantum field structures, J. Phys. A (1998) 231.
20. C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups (Springer-Verlag, Berlin/Heidelberg/New York, 1975).
21. H.-J. Borchers, Algebraic aspects of quantum field theory, in Int. Symp. Math. Probl. Theor. Phys., Kyoto 1975, Lect. Notes Phys. 39 (1975) 283-292.
22. H.-J. Borchers, Algebraic aspects of quantum field theory, in Int. Symp. Math. Probl. Theor. Phys., Kyoto 1975, Lect. Notes Phys. 39 (1975) 283-292.
23. P. Colella and O. E. Lanford, Sample Path Behavior for the Free Markov Field, Lecture Notes Phys. 25 (Springer, Berlin, 1973), p. 44.
24. A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits, LNM 1501 (Springer-Verlag, Berlin/Heidelberg/New York, 1991).
25. C. Deutsch and M. Lavaud, Equilibrium properties of a two-dimensional Coulomb gas, Phys. Rev. A9 (1974) 2598.
26. S.-H. Djah, H. Gottschalk and H. Ouerdiane, Feynman graph representation for the perturbation series for general functional measures, math-ph/0408031, to appear in J. Funct. Anal.
27. R. Fernàndez, J. Fröhlich and A. D. Sokal, Random Walks, Critical Phenomena and triviality in Quantum Field Theory (Springer-Verlag, Berlin/Heidelberg/New York, 1992).
28. J. Fröhlich, Classical and quantum statistical mechanics in one and two dimensions: Two-component Yukawa - and Coulomb systems, Commun. Math. Phys. 47 (1976) 233.
29. J. Fröhlich and Y. M. Park, Remarks on exponential interactions and the quantum sine-Gordon equation in two space-time dimensions, Helv. Phys. Acta 50 (1977) 315.
30. 2= Convergence and perturbation structure, Helv. Phys. Acta 49 (1976) 889.
31. I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions, IV. Some Applications of Harmonic Analysis (Academic Press, New York/London, 1964).
32. H.-O. Georgii, Gibbs measures and phase transitions (W. de Gruyter, Berlin/ New York, 1988).
33. R. Gielerak and P. Ługiewicz, From stochastic differential equation to quantum field theory, Rep. Math. Phys. 44(1/2) (1999) 101.
34. R. Gielerak and P. Ługiewicz, 4D local quantum field theory models from covariant stochastic partial differential equations, Rev. Math. Phys. 13(3) (2001) 335-408.
35. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 2nd edn. (Springer, Berlin/Heidelberg/New York, 1987).
36. 4 Hamiltonian, Fortschr. Phys. 21 (1973) 327.
37. O. W. Greenberg, Generalized free fields and models of local field theory, Ann. Phys. 18 (1969) 158.
38. H. Gottschalk, Particle systems with weakly attractive interaction, SFB 611 preprint Bonn 2002, math-ph/0409029.
39. H. Gottschalk, Wick rotation for holomorphic random fields, in Recent Developments in Stochastic Analysis and Related Topics, World Scientific, Singapore 2004, Proceedings of the First Sino-German Conference on Stochastic Analysis, eds. S. Albeverio, Z.-M. Ma and M. Rckner.
40. M. Grothaus and L. Streit, Construction of relativistic quantum fields in the framework of white noise analysis, J. Math. Phys. 40(11) (1999) 5387.
41. P. R. Halmos, Measure Theory, 2nd edn. (Springer Verlag, Berlin/Heidelberg, 1976).
42. G. E. Johnson, Interacting quantum fields, Rev. Math. Phys. 11(7) (1999) 881, with Erratum ibid. 12 (2000) 687.
43. Yu. Kondratiev and T. Kuna, Correlation functionals for Gibbs measures and Ruelle bounds, Methods Funct. Anal. Topol. 9(1) (2003) 9-58.
44. A. Lenard, Correlation functions and the uniqueness of the state in classical statistical mechanics, Commun. Math. Phys. 30 (1973) 35.
45. K. R. Mecke, Integral geometry in statistical physics, Int. J. Mod. Phys. 12(9) (1998) 861.
46. R. A. Minlos, Generalized Random Processes and their Extension in Measure, Translations in Mathematical Statistics and Probability, Vol. 3 (AMS Providence, 1963), p. 291.
47. G. Morchio and F. Strocchi, Infrared singularities, vacuum structure and pure phases in local quantum field theory, Ann. Inst. H. Poincaré 33 (1980) 251.
48. E. Nelson, Construction of quantum fields from Markoff fields, J. Funct. Anal. 12 (1973) 97.
49. E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973) 211.
50. K. Osterwalder and R. Schrader, Axioms for Euclidean Green's functions I, Commun. Math. Phys. 31 (1973) 83.
51. C. Preston, Random Fields, LNM 534 (Springer, Berlin/Heidelberg/New York, 1976).
52. M. Reed and J. Rosen, Support properties of the free measure for the Boson field, Commun. Math. Phys. 36 (1974) 123.
53. D. Ruelle, Statistical mechanics - rigorous results (Benjamin, London/Amsterdam/ Don Mills (Ontario)/Sydney/Tokyo, 1969).
54. L. A. Santalò, Integral geometry and geometric probability (Addison-Wesley, Reading, 1976).
55. B. Schroer, Infrateilchen in der Quantenfeldtheorie, Fortschr. Phys. 173 (1963) 1527.
56. 2 Euclidean (Quantum) Field Theory (Princeton University Press, Princeton, New Jersey, 1974).
57. D. Stoyan, W. S. Kendall and J. Mecke, Stochastic geometry and it's applications Wiley & Sons, 1987).
58. F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory, Lecture Notes in Physics 51 (World Scientific, Singapore/New York/London/ Hong Kong, 1993).
59. R. F. Streater and A. S. Wightman, PCT, Spin, Statistics and All That (Benjamin, New York, 1964).
60. K. Symanzik, Euclidean quantum field theory, in Local Quantum Theory, ed. R. Jost (Academic Press, New York, 1969).
61. H. Tamura, On the possibility of confinement caused by nonlinear electromagnetic interaction, J. Math. Phys. 32 (1991) 897.
62. M. W. Yoshida, Non-linear continuous maps on abstract Wiener spaces defined on space of tempered distributions, Bull Univ. Electro-Commun. 12 (1999) 101-117.