Non-equilibrium Quantum Field Theory and Entangled Commutation Relations

L. Accardi ; I. Ya. Aref'eva ; I. V. Volovich

Informatics and Automation, Problems of the modern mathematical physics, Tome 228 (2000), pp. 116-135.

Résumé

Non-equilibrium quantum field theory studies the time dependence of processes to which the $S$-matrix approach is inapplicable. One of the new methods of investigation in non-equilibrium quantum theory is the stochastic limit method. This method is an extension of the works by Bogolyubov, Van Hove, and Prigogine and permits the study not only of the system but also of the reservoir degrees of freedom. We consider the stochastic limit of translation invariant Hamiltonians in quantum field theory and show that the master field satisfies a new type of commutation relations, the so-called entangled (or interacting) commutation relations. These relations extend the interacting Fock relations established earlier in non-relativistic QED and the free (or Boltzmann) commutation relations which have been found in the large $N$ limit of QCD. As an application of the stochastic limit method, we consider the photon splitting cascades in magnetic field and show that photons in cascades form entangled states (“triphons”) and obey a new type of statistics corresponding to the entangled commutation relations rather than the Bose statistics.

Bibliographie

[1] Bogolyubov N. N., Shirkov D. V., Vvedenie v teoriyu kvantovannykh polei, Nauka, M., 1973 | MR | Zbl

[2] Bogolyubov N. N., Zadachi dinamicheskoi teorii v statisticheskoi fizike, Gostekhizdat, M., 1946 | MR

[3] Bonch-Bruevich V. L., Tyablikov S. V., Metod funktsii Grina v statisticheskoi mekhanike, Fizmatgiz, M., 1961 | MR

[4] Abrikosov A. A., Gorkov L. P., Dzyaloshinskii I. E., Metody kvantovoi teorii polya v statisticheskoi fizike, Fizmatgiz, M., 1962 | MR | Zbl

[5] Zubarev D. N., Neravnovesnaya statisticheskaya termodinamika, Nauka, M., 1971

[6] Bogolyubov N. N., Tyablikov S. V., “Zapazdyvayuschie i operezhayuschie funktsii Grina v statisticheskoi fizike”, DAN SSSR, 126:1 (1959), 53–56 | Zbl

[7] Schwinger J., “Field theory of unstable particles”, Ann. Phys., 9 (1960), 169–193 | DOI | MR | Zbl

[8] Goldberger M. L., Watson K. M., Collision theory, J. Wiley and Sons Inc., New York–London–Sydney, 1964 | MR

[9] Cohen-Tannoudji C., Dupont-Roc J., Grinberg G., Atom-photon interactions, basic processes and applications, J. Wiley and Sons Inc., New York–London–Sydney, 1992

[10] Feinberg E. L., “Chastitsa s neravnovesnym sobstvennym polem”, Problemy teoreticheskoi fiziki, Pamyati I. E. Tamma, Nauka, M., 1972, 248–264 | MR

[11] Rubakov V. A., Shaposhnikov M. E., “Elektroslaboe nesokhranenie barionnogo chisla v rannei Vselennoi i v stolknoveniyakh chastits pri vysokikh energiyakh”, UFN, 166 (1996), 493–537 | DOI

[12] Walls D. F., Milburn G. J., Quantum optics, Springer-Verl., N. Y. etc., 1994 | MR | Zbl

[13] Prigogine I., Fundamental problems in elementary particle physics: XIV Conseil Phys. Solvay (Brussels, Oct. 1967), Intersci., N. Y., 1968, 100–105

[14] Accardi L., Frigerio A., Lu Y. G., “On the weak coupling limit problem”, Quantum probability and applications, {, IV}, Lect. Notes Math., 1396, 1987, 20–58 | MR

[15] Accardi L., Lu Y. G., Volovich I. V., Quantum theory and its stochastic limit, Springer-Verlag, Berlin, 2002 | MR

[16] Van Hove L. Quantum mechanical perturbations giving rise to a transport equation, Physica, 21 (1955), 517–540 | MR

[17] Faddeev L. D., “O razdelenii effektov samodeistviya i rasseyaniya po teorii vozmuschenii”, DAN SSSR, 152 (1963), 573–576 | MR

[18] Arefeva I. Ya., “Perenormirovannaya teoriya rasseyaniya dlya modeli Yukavy. I: Postroenie odevayuschikh operatorov”, TMF, 14:1 (1973), 3–17

[19] Chew G. F., The analytic $S$-matrix. A basis for nuclear democracy, W. A. Benjamin, Inc., New York–Amsterdam, 1966 | MR

[20] Prigogine I., Henin F., “Kinetic theory and subdynamics”, Problemy teoreticheskoi fiziki, Sb., posvyaschennyi N. N. Bogolyubovu v svyazi s ego shestidesyatiletiem, Nauka, M., 1969, 356–364

[21] Bogolyubov N. N., “Elementarnyi primer ustanovleniya statisticheskogo ravnovesiya v sisteme, svyazannoi s termostatom”, O nekotorykh statisticheskikh metodakh v matematicheskoi fizike, Izd-vo AN USSR, Kiev, 1946, 115–137

[22] Friedrichs K. O., “On the perturbation of continuous spectra”, Commun. Pure and Appl. Math., 1 (1948), 361–406 | DOI | MR | Zbl

[23] Aref'eva I. Ya., “Regge regime in QCD and asymmetric lattice gauge theory”, Phys. Lett. B, 325 (1994), 171 | DOI

[24] Aref'eva I. Ya., Volovich I. V., “Anisotropic asymptotics and high energy scattering in QCD”, Quarks'94, eds. D. Yu. Grigoriev et al., World Sci., Singapore, 1995, 155–170

[25] Matveev V. A., “Origin of the quark counting laws”, Quarks'94, eds. D. Yu. Grigoriev et al., World Sci., Singapore, 1995, 41–51

[26] Accardi L., Kozyrev S. V., Volovich I. V., “Dynamics of dissipative two-level systems in the stochastic approximation”, Phys. Rev. A, 56:4 (1997), 2557–2562 ; arXiv: quant-ph/9706021 | DOI

[27] Accardi L., Kozyrev S. V., Volovich I. V., “On the non-exponential decay in the polaron model”, Phys. Lett. A (to appear)

[28] Aref'eva I. Ya., Volovich I. V., “Quantum group particles and non-archimedean geometry”, Phys. Lett. B, 268 (1991), 179–187 | DOI | MR

[29] Greenberg O. W., “Example of infinite statistics”, Phys. Rev. Lett., 64 (1990), 705–708 | DOI | MR | Zbl

[30] Aref'eva I. Ya., Volovich I. V., “The master field for QCD and $q$-deformed quantum field theory”, Nucl. Phys. B, 462 (1996), 600–615 | DOI | MR

[31] Accardi L., Lu Y. G., “The Wigner semi-circle law in quantum electrodynamics”, Commun. Math. Phys., 180 (1996), 605–632 | DOI | MR | Zbl

[32] Accardi L., Lu Y. G., Volovich I. V., Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations, Publ. IIAS, Kyoto, 1997

[33] Halpern M. B., Schwartz C., The algebras of large $N$ matrix mechanics, preprint, 1998 ; arXiv: hep-th/9809197 | MR

[34] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1981 | MR

[35] Adler S. L., Bahcall J. N., Callan C. G., Rosenbluth M. N., “Photon splitting in a strong magnetic field”, Phys. Rev. Lett., 25 (1970), 1061–1065 | DOI

[36] Adler S. L., “Photon splitting and photon dispersion in a strong magnetic field”, Ann. Phys., 67 (1971), 599–647 | DOI

[37] Berestetskii V. B., Lifshitz E. M., Pitaevskii L. P., Quantum electrodynamics, Pergamon Press, Oxford, 1982

[38] Baring M. G., Harding A. K., Gonthier P. L., The attenuation of gamma-ray emission in strongly-magnetized pulsars, preprint, 1997 ; arXiv: astro-ph/9704210 | Zbl

[39] Harding A. K., Baring M. G., Gonthier P. L., Photon splitting cascades in gamma-ray pulsars and the spectrum of PSR1509-58, preprint, 1996; arXiv: astro-ph/9609167

[40] Baring M. G., “Photon-splitting limits to the hardness of emission in strongly magnetized soft gamma repeaters”, Astrophys. J., 440 (1995), L69–L72 | DOI

[41] Harding A. K., Baring M. G., Photon splitting in soft gamma repeaters, preprint, 1996; arXiv: astro-ph/9603095

[42] Protheroe R. J., Origin and propagation of the highest energy cosmic rays, preprint, 1996; arXiv: astro-ph/9610100 | Zbl

[43] Adler S. L., Schubert C., Photon splitting in a strong magnetic field: recalculation and comparison with previous calculations, preprint, 1996; arXiv: hep-th/9605035

[44] Baier V. N., Milstein A. I., Shaisultanov R. Zh., “Photon splitting in a very strong magnetic field”, Phys. Rev. Lett., 77 (1996), 1691–1694 | DOI

[45] Heyl J. S., Hernquist L., Birefringence and dichroism of the QED vacuum, preprint, 1997 ; arXiv: hep-ph/9705367 | MR