Analytic continuation of thermal <i>N</i>-point functions from imaginary to real energies

R. Baier; A. Niégawa

doi:10.1103/PhysRevD.49.4107

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Analytic continuation of thermal N-point functions from imaginary to real energies

R. Baier

A. Niégawa

Fakultät für Physik, Universität Bielefeld, 33501 Bielefeld, Germany

Department of Physics, Osaka City University, Sumiyoshi-ku, Osaka 558, Japan

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Phys. Rev. D 49, 4107 – Published 15 April, 1994

DOI: https://doi.org/10.1103/PhysRevD.49.4107

Abstract

Thermal n-point Green functions in the framework of quantum field theory at finite temperature are considered. We reanalyze how analytic continuations from imaginary to real energies relate these functions originally defined in the imaginary-time formalism to retarded and advanced realtime ones. We described a new and rather simple method which is valid to all orders of perturbation theory and which has the further advantage that it is independent of approximations often applied in actual finite-order calculations.

References (18)

T. Matsubara, Prog. Theor. Phys. 14, 351 (1955).
Y. Takahashi and H. Umezawa, Collect. Phenom. 2, 55 (1975); A. J. Niemi and G. W. Semenoff, Ann. Phys. (N.Y.) 152, 105 (1984); Nucl Phys. B230, 181 (1984).
J. Schwinger, J. Math. Phys. 2, 407 (1961); L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)]. See also, R. A. Craig, J. Math. Phys. 9, 605 (1968).
A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Pergamon, Oxford, 1965); H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo Field Dynamics and Condensed States (North Holland, Amsterdam, 1982); J. I. Kapusta, Finite Temperature Field Theory (Cambridge University Press, Cambridge, England, 1989).
R. Mills, Propagators for Many Particle Systems (Gordon and Breach, New York, 1969).
K. C. Chou, Z. B. Su, B. L. Hao and L. Yu, Phys. Rep. 118, 1 (1985).
N. P. Landsman and Ch. G. van Weert, Phys. Rep. 145, 141 (1987).
G. Baym and N. D. Mermin, J. Math. Phys. 2, 232 (1961).
E. Braaten and R. D. Pisarski, Nucl. Phys. B337, 569 (1990); J. Frenkel and J. C. Taylor, ibid. 334, 199 (1990).
R. Baier, B. Pire and D. Schiff, Phys. Lett. B 238, 367 (1990); Z. Phys. C 51, 581 (1991) H. Nakkagawa, A. Niégawa and H. Yokota, Phys. Lett. B 244, 63 (1990); ibid. 273, 549(E) (1991); N. Ashida, A. Niégawa, H. Nakkagawa and H. Yokota, Phys. Rev. D 44, 473 (1991); ibid. 45, 1432(E) (1992). Earlier work is quoted in these papers.
T. S. Evans, Phys. Lett. B 249, 286 (1990); ibid. 252, 108 (1990).
R. Kobes, Phys. Rev. D 42, 562 (1990); ibid. 43, 1269 (1991); Phys. Rev. Lett. 67, 1384 (1991); P. Aurenche, E. Petitgirard, and T. del Rio Gaztelurrutia, Phys. Lett. B 297, 337 (1992).
T. S. Evans, Nucl. Phys. B374, 340 (1992).
P. Aurenche and T. Becherrawy, Nucl. Phys. B379, 259 (1992); J. C. Taylor, Phys. Rev. D 47, 725 (1993). See also, M. A. van Eijck and Ch. G. van Weert, Phys. Lett. B 277, 305 (1992); F Guerin, Institut Non Linéaire de Nice Report No. INLN 1993/14 (unpublished).
Strictly speaking, we choose Im $t_{I}$ to be a positive infinitesimal number.
R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957); P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959).
A. Niégawa, Phys. Rev. D 40, 1199 (1989).
T. Evans, Phys. Rev. D 47, R4196 (1993); Imperial College Report No. Imperial/TP/93 94/4, 1993 (unpublished).

[c1] T. Matsubara, Prog. Theor. Phys. 14, 351 (1955).

[c2] Y. Takahashi and H. Umezawa, Collect. Phenom. 2, 55 (1975); A. J. Niemi and G. W. Semenoff, Ann. Phys. (N.Y.) 152, 105 (1984); Nucl Phys. B230, 181 (1984).

[c3] J. Schwinger, J. Math. Phys. 2, 407 (1961); L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)]. See also, R. A. Craig, J. Math. Phys. 9, 605 (1968).

[c4] A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Pergamon, Oxford, 1965); H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo Field Dynamics and Condensed States (North Holland, Amsterdam, 1982); J. I. Kapusta, Finite Temperature Field Theory (Cambridge University Press, Cambridge, England, 1989).

[c5] R. Mills, Propagators for Many Particle Systems (Gordon and Breach, New York, 1969).

[c6] K. C. Chou, Z. B. Su, B. L. Hao and L. Yu, Phys. Rep. 118, 1 (1985).

[c7] N. P. Landsman and Ch. G. van Weert, Phys. Rep. 145, 141 (1987).

[c8] G. Baym and N. D. Mermin, J. Math. Phys. 2, 232 (1961).

[c9] E. Braaten and R. D. Pisarski, Nucl. Phys. B337, 569 (1990); J. Frenkel and J. C. Taylor, ibid. 334, 199 (1990).

[c10] R. Baier, B. Pire and D. Schiff, Phys. Lett. B 238, 367 (1990); Z. Phys. C 51, 581 (1991) H. Nakkagawa, A. Niégawa and H. Yokota, Phys. Lett. B 244, 63 (1990); ibid. 273, 549(E) (1991); N. Ashida, A. Niégawa, H. Nakkagawa and H. Yokota, Phys. Rev. D 44, 473 (1991); ibid. 45, 1432(E) (1992). Earlier work is quoted in these papers.

[c11] T. S. Evans, Phys. Lett. B 249, 286 (1990); ibid. 252, 108 (1990).

[c12] R. Kobes, Phys. Rev. D 42, 562 (1990); ibid. 43, 1269 (1991); Phys. Rev. Lett. 67, 1384 (1991); P. Aurenche, E. Petitgirard, and T. del Rio Gaztelurrutia, Phys. Lett. B 297, 337 (1992).

[c13] T. S. Evans, Nucl. Phys. B374, 340 (1992).

[c14] P. Aurenche and T. Becherrawy, Nucl. Phys. B379, 259 (1992); J. C. Taylor, Phys. Rev. D 47, 725 (1993). See also, M. A. van Eijck and Ch. G. van Weert, Phys. Lett. B 277, 305 (1992); F Guerin, Institut Non Linéaire de Nice Report No. INLN 1993/14 (unpublished).

[c15] Strictly speaking, we choose Im $t_{I}$ to be a positive infinitesimal number.

[c16] R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957); P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959).

[c17] A. Niégawa, Phys. Rev. D 40, 1199 (1989).

[c18] T. Evans, Phys. Rev. D 47, R4196 (1993); Imperial College Report No. Imperial/TP/93 94/4, 1993 (unpublished).

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Analytic continuation of thermal N-point functions from imaginary to real energies

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