Functional methods and perturbation theory

J. Iliopoulos; C. Itzykson; A. Martin

doi:10.1103/RevModPhys.47.165

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Functional methods and perturbation theory

J. Iliopoulos, C. Itzykson, and A. Martin

J. Iliopoulos

Laboratoire de Physique Théorique et Hautes Energies, Orsay, France

C. Itzykson

Service de Physique Théorique, Centre d' Etudes Nucléaires de Saclay, 91190 Gif-sur-Yvette, France

A. Martin

Division Théorique, CERN, Geneva

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Rev. Mod. Phys. 47, 165 – Published 1 January, 1975

DOI: https://doi.org/10.1103/RevModPhys.47.165

Abstract

The framework of functional integrals in field theory is convenient for presenting a unified view of the perturbation expansion according to the number of loops. We review the calculation of the generating functional for irreducible Green functions and its renormalization properties. Calculations of the effective potential and $Z$ -function are carried up to the order of two loops for the self-coupled scalar field. This is applied to compute the coefficients of the Callan-Symanzik equation which describes the short distance behavior of the theory. In conclusion, we present some speculations concerning the positivity of the coupling constant, and its relation to the on -shell two-particle scattering amplitude.

References (23)

Adler, S., 1973, Phys. Rev. D 8, 2400
Atkinson, D., 1968, Nucl. Phys. B 7, 375
Bonnier, B., and R. Vinh-Mau, 1968, Phys. Rev. 165, 1923
Callan, C. G., 1970, Phys. Rev. D 2, 1541
Callan, C. G., 1973, lectures at Cargèse, 1973
Coleman, S., 1973, "Secret Symmetry," Ettore Majorana Summer School Proceedings (Academic, New York)
Coleman, S., and E. Weinberg, 1973, Phys. Rev. D 7, 1888
Gell-Mann, M., and F. E. Low, 1954, Phys. Rev. 95, 1300
Grassberger, P., and H. Kuhnelt, 1973, University of Bonn preprint
Jackiw, R., 1974, Phys. Rev. D 9, 1686
Jin, Y. S., and A. Martin, 1964, Phys. Rev. 135 B, 1369
Jona-Lasinio, G., 1964, Nuovo Cimento 34, 1790
Lowenstein, J. H., 1971, Commun. Math. Phys. 24, 1
Laukaszuk, L., and A. Martin, 1967, Nuovo Cimento 52, 122
Martin, A., 1965, in High Energy Physics and Elementary Particles, lectures presented at a Seminar at Trieste, 1965 (IAEA, Vienna) p. 155
Nambu, Y., 1968, Phys. Lett. 26 B, 626
Ovsiannikov, L. V., 1956, Dokl. Akad. Nauk. SSSR 109, 1121
Schwinger, J., 1951, Phys. Rev. 82, 664
Stüeckelberg, E. C. G., and A. Petermann, 1953, Helv. Phys. Acta 26, 499
Symanzik, K., 1970, Commun. Math. Phys. 16, 48
Symanzik, K., 1970, Commun. Math. Phys. 18, 227
Wilson, K., and J. Kogut, 1974, Phys. Report 12C, 2
Zinn-Justin, J., 1973, "Wilson's theory of critical phenomena and renormalized perturbation theory," lectures at Cargèse, 1973

[c1] Adler, S., 1973, Phys. Rev. D 8, 2400

[c2] Atkinson, D., 1968, Nucl. Phys. B 7, 375

[c3] Bonnier, B., and R. Vinh-Mau, 1968, Phys. Rev. 165, 1923

[c4] Callan, C. G., 1970, Phys. Rev. D 2, 1541

[c5] Callan, C. G., 1973, lectures at Cargèse, 1973

[c6] Coleman, S., 1973, "Secret Symmetry," Ettore Majorana Summer School Proceedings (Academic, New York)

[c7] Coleman, S., and E. Weinberg, 1973, Phys. Rev. D 7, 1888

[c8] Gell-Mann, M., and F. E. Low, 1954, Phys. Rev. 95, 1300

[c9] Grassberger, P., and H. Kuhnelt, 1973, University of Bonn preprint

[c10] Jackiw, R., 1974, Phys. Rev. D 9, 1686

[c11] Jin, Y. S., and A. Martin, 1964, Phys. Rev. 135 B, 1369

[c12] Jona-Lasinio, G., 1964, Nuovo Cimento 34, 1790

[c13] Lowenstein, J. H., 1971, Commun. Math. Phys. 24, 1

[c14] Laukaszuk, L., and A. Martin, 1967, Nuovo Cimento 52, 122

[c15] Martin, A., 1965, in High Energy Physics and Elementary Particles, lectures presented at a Seminar at Trieste, 1965 (IAEA, Vienna) p. 155

[c16] Nambu, Y., 1968, Phys. Lett. 26 B, 626

[c17] Ovsiannikov, L. V., 1956, Dokl. Akad. Nauk. SSSR 109, 1121

[c18] Schwinger, J., 1951, Phys. Rev. 82, 664

[c19] Stüeckelberg, E. C. G., and A. Petermann, 1953, Helv. Phys. Acta 26, 499

[c20] Symanzik, K., 1970, Commun. Math. Phys. 16, 48

[c21] Symanzik, K., 1970, Commun. Math. Phys. 18, 227

[c22] Wilson, K., and J. Kogut, 1974, Phys. Report 12C, 2

[c23] Zinn-Justin, J., 1973, "Wilson's theory of critical phenomena and renormalized perturbation theory," lectures at Cargèse, 1973

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Functional methods and perturbation theory

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