Equal-Time Commutators and Electromagnetic Mass Splittings

David G. Boulware; S. Deser

doi:10.1103/PhysRev.175.1912

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Equal-Time Commutators and Electromagnetic Mass Splittings

David G. Boulware^*,†

S. Deser^‡

International Centre for Theoretical Physics, Trieste, Italy

Physics Department, Brandeis University, Waltham, Massachusetts 02154

^*On leave of absence from Physics Department, University of Washington, Seattle, Wash.
^†Work supported by U. S. Atomic Energy Commission under Grant No. RCO-1388B.
^‡Work supported by U. S. Air Force Office of Aerospace Research under Office of Scientific Research Grant No. AF 368-67.

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Phys. Rev. 175, 1912 – Published 25 November, 1968

DOI: https://doi.org/10.1103/PhysRev.175.1912

Abstract

The relation between electromagnetic mass shifts (or splittings) to order $α$ and equal-time commutators is analyzed. If the invariants in the forward spin-averaged Compton amplitude go as $k^{- 2}$ for $k^{2} \to \infty$ , it is possible to relate the quadratic and logarithmic divergences in $δ m^{2}$ to certain Lorentz scalars formed from the commutators. These results are illustrated by models with currents bilinear or linear in canonical fields. The Jost-Lehmann representation is then used to analyze more general behavior. It is shown that the spectral function may generate a higher-rank tensor character of the amplitude than that expected from its kinematic factors. The relation is then more complicated even if the commutators still exist. Finally, detailed criteria for the existence of equal-time commutators are developed.

References (23)

J. D. Bjorken, Phys. Rev. 148, 1467 (1966)
R. Jost and H. Lehmann, Nuovo Cimento 5, 1598 (1957)
A. S. Wightman, Phys. Rev. 101, 860 (1956)
R. A. Brandt and J. Sucher, Phys. Rev. Letters 20, 1131 (1968) to be published
S. Deser, Copenhagen Summer Institute Lecture 1967 (SINBI Proceedings, Copenhagen, 1967) M. Halpern and G. Segrè, Phys. Rev. Letters 19, 611 (1967) ibid.19, 1000 (E) (1967) G. C. Wick and B. Zumino, Phys. Letters 25, 479 (1967)
L. S. Brown, Phys. Rev. 150, 1338 (1966)
W. N. Cottingham, Ann. Phys. (N.Y.) 25, 424 (1963)
Omitted endnote
K. A. Johnson and F. E. Low, Progr. Theoret. Phys. Suppl. 37 & 38, 74 (1966)
N. Kroll, T. D. Lee, and B. Zumino, Phys. Rev. 157, 1376 (1967)
T. D. Lee, Phys. Rev. 168, 1714 (1968)
T. Das et al., Phys. Rev. Letters 18, 759 (1967)
S. Weinberg, Phys. Rev. Letters 18, 507 (1967)
Omitted endnote
W. Zimmerman, New York University report (unpublished)
Omitted endnote
Omitted endnote
H. Harari, Phys. Rev. Letters 17, 1303 (1966) W. N. Cottingham and J. Gibb, ibid. 18, 833 (1967)
B. Schroer and P. Stichel, Commun. Math. Phys. 3, 258 (1966)
Bateman Manuscript Project, edited by A. Erdelyi (McGrawHill Book Co., New York, 1954), Vol. II, p. 128 (8) G. N. Watson, Theory of Bessel Functions (Cambridge University Press, London, 1958), 2nd ed., p. 348
D. G. Boulware and S. Deser, J. Math. Phys. 8, 1468 (1967) S. Berman and Y. Frishman, Phys. Rev. 105, 1555 (1968)
Omitted endnote
H. Epstein and R. Jackiw (unpublished)

[c1] J. D. Bjorken, Phys. Rev. 148, 1467 (1966)

[c2] R. Jost and H. Lehmann, Nuovo Cimento 5, 1598 (1957)

[c3] A. S. Wightman, Phys. Rev. 101, 860 (1956)

[c4] R. A. Brandt and J. Sucher, Phys. Rev. Letters 20, 1131 (1968) to be published

[c5] S. Deser, Copenhagen Summer Institute Lecture 1967 (SINBI Proceedings, Copenhagen, 1967) M. Halpern and G. Segrè, Phys. Rev. Letters 19, 611 (1967) ibid.19, 1000 (E) (1967) G. C. Wick and B. Zumino, Phys. Letters 25, 479 (1967)

[c6] L. S. Brown, Phys. Rev. 150, 1338 (1966)

[c7] W. N. Cottingham, Ann. Phys. (N.Y.) 25, 424 (1963)

[c8] Omitted endnote

[c9] K. A. Johnson and F. E. Low, Progr. Theoret. Phys. Suppl. 37 & 38, 74 (1966)

[c10] N. Kroll, T. D. Lee, and B. Zumino, Phys. Rev. 157, 1376 (1967)

[c11] T. D. Lee, Phys. Rev. 168, 1714 (1968)

[c12] T. Das et al., Phys. Rev. Letters 18, 759 (1967)

[c13] S. Weinberg, Phys. Rev. Letters 18, 507 (1967)

[c14] Omitted endnote

[c15] W. Zimmerman, New York University report (unpublished)

[c16] Omitted endnote

[c17] Omitted endnote

[c18] H. Harari, Phys. Rev. Letters 17, 1303 (1966) W. N. Cottingham and J. Gibb, ibid. 18, 833 (1967)

[c19] B. Schroer and P. Stichel, Commun. Math. Phys. 3, 258 (1966)

[c20] Bateman Manuscript Project, edited by A. Erdelyi (McGrawHill Book Co., New York, 1954), Vol. II, p. 128 (8) G. N. Watson, Theory of Bessel Functions (Cambridge University Press, London, 1958), 2nd ed., p. 348

[c21] D. G. Boulware and S. Deser, J. Math. Phys. 8, 1468 (1967) S. Berman and Y. Frishman, Phys. Rev. 105, 1555 (1968)

[c22] Omitted endnote

[c23] H. Epstein and R. Jackiw (unpublished)

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Equal-Time Commutators and Electromagnetic Mass Splittings

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