On the Calculation of Self-Energies in Quantum Theory by Analytic Continuation

S. Bertil Nilsson

doi:10.1103/PhysRev.73.903

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On the Calculation of Self-Energies in Quantum Theory by Analytic Continuation

S. Bertil Nilsson

Institute of Mechanics and Mathematical Physics, University of Lund, Sweden

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Phys. Rev. 73, 903 – Published 15 April, 1948

DOI: https://doi.org/10.1103/PhysRev.73.903

Abstract

Riesz's method of solving hyperbolic differential equations by analytic continuation has been used by Gustafson to eliminate infinities in quantum theory. Treating the one-electron case, he found finite values of the self-energy integrals in the second approximation, also for those integrals for which the $λ$ -limiting process fails (without the further assumption of negative-energy photons). In the present paper it is shown that the general result of Gustafson's procedure is to remove all divergences normally appearing in self-energy expressions, except logarithmic divergences. Thus the total self-energy of the electron, to the second approximation, is found to be zero on the one-electron theory if calculated by this method, whereas in the hole theory the logarithmically divergent expression of Weisskopf is retained. A proposal by Pauli to alter the commutation rules in a certain way gives essentially the same results.

References (15)

N. E. Fremberg, Fysiogr. Sällsk. i Lund Förh. 15, No. 27 (1945)
N. E. Fremberg, Proc. Roy. Soc. A188, 18 (1946)
N. E. Fremberg, "A Study of Generalized Hyperbolic Potentials," Medd. Lunds Univ. Mat. Sem. 7 (1946), thesis
T. Gustafson, Fysiogr. Sällsk. i Lund Förh. 15, No. 28 (1945)
T. Gustafson, Fysiogr. Sällsk. i Lund Förh. 16, No. 2 (1946)
T. Gustafson, Nature 157, No. 3996, 734 (1946)
T. Gustafson, Nature 158, No. 4008, 273 (1946)
T. Gustafson, Arkiv f. Mat., Astr. o. Fys. 34A, No. 2 (1947)
S. B. Nilsson, Fysiogr. Sällsk. i Lund Förh. 16, No. 24 (1946)
W. Pauli, Rev. Mod. Phys. 15, 175 (1943)
M. Riesz, C. R. du Congrès internat. d. math. (Oslo, 1936) II, p. 44
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I. Waller, Zeits. f. Physik 62, 673 (1930)
V. F. Weisskopf, Phys. Rev. 56, 72 (1939)
S. T. Ma, Phys. Rev. 71, 787 (1947)

[c1] N. E. Fremberg, Fysiogr. Sällsk. i Lund Förh. 15, No. 27 (1945)

[c2] N. E. Fremberg, Proc. Roy. Soc. A188, 18 (1946)

[c3] N. E. Fremberg, "A Study of Generalized Hyperbolic Potentials," Medd. Lunds Univ. Mat. Sem. 7 (1946), thesis

[c4] T. Gustafson, Fysiogr. Sällsk. i Lund Förh. 15, No. 28 (1945)

[c5] T. Gustafson, Fysiogr. Sällsk. i Lund Förh. 16, No. 2 (1946)

[c6] T. Gustafson, Nature 157, No. 3996, 734 (1946)

[c7] T. Gustafson, Nature 158, No. 4008, 273 (1946)

[c8] T. Gustafson, Arkiv f. Mat., Astr. o. Fys. 34A, No. 2 (1947)

[c9] S. B. Nilsson, Fysiogr. Sällsk. i Lund Förh. 16, No. 24 (1946)

[c10] W. Pauli, Rev. Mod. Phys. 15, 175 (1943)

[c11] M. Riesz, C. R. du Congrès internat. d. math. (Oslo, 1936) II, p. 44

[c12] M. Riesz, Conférence à la Réunion internat. d. Math. tenue à Paris en Juillet 1937, p. 153 (1939)

[c13] I. Waller, Zeits. f. Physik 62, 673 (1930)

[c14] V. F. Weisskopf, Phys. Rev. 56, 72 (1939)

[c15] S. T. Ma, Phys. Rev. 71, 787 (1947)

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On the Calculation of Self-Energies in Quantum Theory by Analytic Continuation

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